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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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25
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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