Learning Theory and Algorithms for Second-Price Auctions with Reserve

MEHRYAR MOHRI MOHRI@ COURANT INSTITUTE & GOOGLE RESEARCH..

Joint work with Andrés Muñoz Medina

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AuctionsStandard method for buying or selling goods:

• U.S. government: Treasury bills.

• Christie’s or Sotheby’s: art.

• eBay: everything, e.g., ‘honeymoon wife replacement’.

• search engine companies: advertising rights.

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AuctionsInteraction between buyers and sellers:

game-theoretical analysis.

• mechanism design.

• study of properties.

This talk:

learning theory analysis.

• repeated auctions.

• leveraging data.

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Some Auction TypesEnglish auctions: interactive format; seller gradually increases the price until a single bidder is left.

Dutch auctions (flowers in the Netherlands): interactive format; seller gradually decreases the price until some bidder accepts to pay.

First-price sealed-bid auctions (e.g. NYC apartments): non-interactive; simultaneous bids, highest bidder wins and pays the value of his bid.

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Second-Price Auctionsaka Vickrey auctions: e.g., eBay.

• bidders submit bids simultaneously.

• highest bidder wins and pays the value of the second-highest bid.

• truthful bidding is a dominated strategy.

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(William Vickrey, 1961)

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TruthfulnessBidder with value , other bids fixed.

• if : change only if bidder wins and wasn’t before and second-highest bid is ; payoff is

• if : change only if bidder loses and used to win. and second-highest bid ; payoff was

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i vi

vi�bj0.

vi�bj�0.

bi > vi

bi < vi

bj 2 [vi, bi]

bj 2 [bi, vi]

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SPA with ReserveSecond-price auctions with reserve: e.g.., Ad Exchanges.

• seller announces a reserve price and,

• bidders submit bids simultaneously.

• winning bidder (if any) wins and pays the maximum of the value of the second-highest bid and .

• truthful bidding is a dominated strategy.

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r

r

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ExampleSuppose the seller’s value is and there is a single bidder whose value is uniformly distributed over .

• no reserve price: item sold at value .

• reserve price: how should it be chosen?

• probability for bid being above .

• expected revenue , thus is optimal.

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(1�r) r

r(1�r) r= 12

[0, 1]

0

0

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Ad Exchanges

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Ad ExchangesSignificant fraction of the revenue of search engine and popular online sites:

• Microsoft, Yahoo!, Google, OpenX, AppNexus.

• Multi-billion dollar industry.

Choice of reserve price:

• main mechanism trough which the auction revenue can be influenced.

• if set too low, winner may end up paying too little; if set too high, the ad slot could be lost.

how can we select the reserve price to optimize revenue?

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This TalkLearning formulation.

Theoretical guarantees.

Algorithms.

Experimental results.

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Previous ML WorkIncentive compatible auctions (Balcan et al., 2008; Blum et al., 2004).

Predicting bid landscapes (Cui et al., 2011).

Revenue optimization for sponsored ads (Zhue et al., 2009; He et

al., 2013; Devanur & Kakade, 2009).

Bandit setting with no feature (Cesa-Bianchi et al., 2013; see also

Ostrovsky & Schwarz, 2011).

Strategic regret minimization (Amin et al., 2013; Munoz & MM, 2014).

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Loss FunctionAuction revenue can be defined in terms of the pair of highest bids :

Equivalently, loss define by

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b = (b(1), b(2))

Rev(r,b) = b(2)1r<b(2) + r 1b(2)rb(1) .

L(r,b) = �Rev(r,b).

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Learning Formulation : public information about auction (features).

: bid space.

: hypothesis set.

distribution over .

Problem: find with small generalization error,

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x 2 X ✓ RN

B ✓ R2+

H ✓ RX

D X ⇥ B

h 2 H

E(x,b)⇠D

[L(h(x),b)].

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Loss FunctionProperties:

• discontinuous.

• non-differentiable.

• non-convex.

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0 1 2 3 4 5 6 7-5

-4

-3

-2

-1

0

b(1)

b(2)

−b(2)

Can we derive guarantees for learning with this loss function?

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Loss Decomposition

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0 1 2 3 4 5 6 7-5

-4

-3

-2

-1

0

b(1)

b(2)

−b(2)

0 1 2 3 4 5 6 7-5

-4

-3

-2

-1

0

b(1)

b(2)

−b(2)

0 1 2 3 4 5 6 7-1

0

1

2

3

4

b(1)0 1 2 3 4 5 6 7

-5

-4

-3

-2

-1

0

b(1)

b(2)

−b(2)

0 1 2 3 4 5 6 7-1

0

1

2

3

4

b(1)

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Generalization BoundTheorem: let and let be a hypothesis set with pseudo-dimension . Then, for any , with probability over the choice of a sample of size ,

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HM = supb2B b(1)

d = Pdim(H) �>01�� S m

L(h) bLS(h) + 2Rm(H) + 2M

r2d log em

d

m+M

slog

1�

2m.

Can we design algorithms minimizing the right-hand side?

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This TalkLearning formulation.

Theoretical guarantees.

Algorithms.

Experimental results.

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No Feature CaseProblem: find optimal reserve price,

Algorithm:

• optimum one of highest bids.

• naive in .

• sorting solution in .

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minr2R

nX

i=1

L(r,bi).

O(m2)

O(m logm)

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Convex Surrogate Loss

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Convex Surrogate LossNo useful convex surrogate loss.

Theorem: Let be a bounded function, convex with respect to its first argument. If is consistent with , then is constant for every .

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Lc : [0,M ]⇥ [0,M ] ! RLc

(r, b) 7! �r1rb Lc(·, b)b 2 [0,M ]

Which loss function should we use?

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Continuous Surrogate Loss

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0 1 2 3 4 5

-4

-3

-2

-1

0

1b2 (1 + γ)b1

b1

L�loss function

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Consistency ResultsTheorem: let and let be a closed convex subset of a linear space of functions containing . Then,

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HM = supb2B b(1)

0

L(h⇤) L(h⇤�) L�(h

⇤�) + �M.

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Learning GuaranteesTheorem: fix . Then, for any , with probability at least over the choice of a sample of size ,

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� 2 (0, 1] �>01�� S m

L�(h) bL�,S(h) +2

�Rm(H) +M

slog

1�

2m.

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AlgorithmOptimization problem: for fixed .

• difficulty: optimizing sum of non-convex functions.

• solution: DC-programming (Difference of Convex).

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minkwk⇤

mX

i=1

L�(w · xi,bi).

� 2 (0, 1]

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Difference of Convex Functions

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L� L� = u� v

u

v

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DC-ProgrammingConvex-concave procedure: replace at iteration with upper bound

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F (w) = f(w)� g(w)(t+ 1)

(Tao and Hoai, 1997; Yuille and Rangarajan, 2002)

bF (w) = f(w)� g(wt)� �g(wt) · (w �wt),

with �g(wt) 2 @g(wt).

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Algorithm

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SecondPriceReserve()

1 w w0

2 for t 1 to T do

3 v DCA(wt�1)4 u v

kvk5 ⌘⇤ min0⌘⇤

Pu·xi>0 L�(⌘u · xi,bi)

6 wt ⌘⇤v7 return w

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Line SearchObservation: is positive homogenous, for all ,

Consequence: line search equivalent to no-feature minimization algorithm; for

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L�

L�(⌘r, ⌘b) = ⌘L�(r,b).

⌘>0

w0>xi>0

mX

i=1

L�(⌘w0>xi,bi) =

mX

i=1

�w0

>xi

�L�

✓⌘,

bi

w0>xi

◆.

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This TalkLearning formulation.

Theoretical guarantees.

Algorithms.

Experimental results.

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Experimental Results

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-0.2-0.15

-0.1-0.05

0 0.05

0.1 0.15

0.2

200 300 400 600 800 1000120016002400

DC CVX NF

-0.2-0.1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

100 200 400 800 1600

NF CVX DC Reg

-0.15-0.1

-0.05 0

0.05 0.1

0.15 0.2

0.25 0.3

0.35 0.4

100 200 400 800 1600

NF CVX DC Reg

-0.2-0.1

0 0.1 0.2 0.3 0.4 0.5

100 200 400 800 1600

NF CVX DC Reg

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Distribution of Reserve Prices

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0

200

400

600

800

1000

1200

0 10Reserve

DCCVXReg

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eBay Sport-Card Data Set

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30

35

40

45

50

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CVX NF DC HB NR

Rev

enue

Data: http://cims.nyu.edu/~munoz/data. Random 2000 training pts - 2000 test pts.

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ConclusionTheory, algorithms, and experiments for second-price auctions with reserve.

• scaling up DC algorithm.

• study of dependencies.

• effect of using revenue optimization algorithm.

• better initialization.

Learning and auctions:

• many other scenarios and types of auctions.

• Example: Generalized Second-Price auctions (GSPs).

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