Competitive Auctions and Digital Goods

Andrew Goldberg, Jason Hartline,

and Andrew Wright

presenting: Keren Horowitz,

Ziv Yirmeyahu

Introduction

• Selling a large number of items

• every customer wants a single item

• unlimited supply

• examples:– downloadable software– pay-per-view movies

Motivation

• Mechanism Design purpose:– to maximize seller’s revenues!

• Using:– Fixed price auctions– Multiple price auctions– Deterministic auctions– Randomized auctions– All single round, sealed bids

The economic view

0

10

20

30

40

50

60

Demand

Quantity

Price

The economic view

0

10

20

30

40

50

60

Demand

Revenues = Price X Quantity

Quantity

Price

Notations

• Set of Bids, B= , sorted: ,

• so

• = utility of bidder I

• T =

• F = Optimal fixed price revenues

• R = current game revenues

• assumption: h is small compare to F

nbb ...1 1 ii bb

hnl bbbb &1

iu

ni iu

1

Notations 2

• Truthful auction: encourage bidders to bid their utility value

• Competitive auction: yields revenues within a constant factor of optimal fixed price.

• We use computer science analysis similar to online algorithms analysis where we check if 1

F

R

For example:

• The k-item vickery auction: – truthful, single price

• Worst case scenario:– k bidders bid at h, n-k bidders bid at 1.

• Vickery revenues (R) = k

• optimal fixed price revenues (F)

•

hk

hF

R 1

How T relates to F

• Theorem: T/(2log h) F

• First, see that F uses the price opt(B):)1(maxarg)( inbBopt iBbi

How T relates to F (2)

• Proof:• 1. Divide the bids into log h bins.

• 2. the sum of the bids = T => there exists a bin, say X, that sums to at least T/(log h).

• 3. the lowest bid in X is at least 1/2 the highest.

• 4. choose price(B) to be the lowest bid in that bin.

• 5. Each bid in that bin will contribute at least half its value.

• 6. h

Tbb i

binbi

binb iilog2

1

2

1

2

1

2

4

16

X

Random Sampling Auctions

• Select a subset B’ of B (the set of bids),

• Compute f(B’) to be the optimal fixed price for B’

• Use f(B’) as a threshold to B\B’

• Bidder I wins the auction at price f(B’) if

mB

)'(Bfbi

Performance analysis

• Theorem: assume , then with a probability

• Proof is basing in the Chernoff bound.

• By this Theorem:

72/36/ 401 aa ee

1F

R

Fah 6/FR

Weighted Pairing Auction

• A bid-independent, truthful, multi price auction:

BbBf )( Bbb

bPW

''

..

Bids

distribution

Weighted pairing performance

• if , then for the weighted pairing auction E(R)=(T/(log h)).

• But still, weighted pairing isn’t competitive, in the worst case, it has a tight bound of:

)log/( hF

Fh 4

An upper bound

• We can see that no truthful auction, even a multi price one, can do better then F in the average case:

• Theorem: for any truthful auction: FRE ][

An upper bound - proof

• Let’s define:

– the probability a bid I is satisfy

– expected cost to winning bidder I

– expected profit (gain) for bidder I

• (1)

ip

ic

ig)( iiii cupg

An upper bound - proof(2)

• Lemma: in a truthful auction:

• Proof:• if , then • if , then• together: • since , we have .

jiji ppbb

iji buu

jji buu )()( jijiii cbpcbp

)()( jjjiji cbpcbp

)()( ijiijj bbpbbp

ji bb ji pp

An upper bound - proof(3)

• Theorem: for any truthful auction:

• Proof:• if bidder I-1 had the utility value of his gain will

not exceed: (2)

• so:

ib

FRE )(

)( 11 iiii cbpg

)()( 11111 iiiiii cbpbbp

)( 1111 iiiiii cbbbpg

111 )( iiii gbbp

An upper bound - proof(4)

• We can recursively extend in the same way until:

• define the total expected revenue from bidder I as:

• rewrite (1) as:

• using equation (3) we get:

iii cpR

1

11 )(

i

jjjji bbpg

)( iiii cupg iiii gbpR

)3(

1

11 )(

i

jjjjiii bbpbpR

An upper bound - proof(5)

n

i

i

jjjj

n

iii bbpbp

1

1

11

1

)(

1

1

][i

jiRRE

1

11

1

))((n

ijjj

n

iii jnbbpbp

1

11

1

1

))((n

jjjj

n

jjjnn jnbbpbpbp

1

11 )()1(][

n

ijjjnn jnbjnbpbpRE

1

11 ))((

n

jjjjjnn jnbbbpbp

)1( jnbV jj

jb

1

11 )()1(][

n

ijjjnn jnbjnbpbpRE

An upper bound - proof(6)

• define , this is also the revenues from a fixed price auction using as the price.

• Note that .

• Rearrange the sum over and define :

FV j

1

11][

n

ijjjnn VVpVpRE

jV 00 p

n

ijjj VppRE

11)(][

An upper bound - proof(7)

• but and by the lemma in the beginning we showed that so:

• this sum telescopes to but so:

n

ijjj VppRE

11)(][

00 p0ppn

FV j 01 jj pp

n

ijj ppFRE

11)(][

FFpRE n ][

Deterministic Optimal Threshold Auction

• A bid-independent auction, meaning bidder i‘s bid value should only determine whether bidder i wins or losses. The bid value should not determine bidder i’s price.

• Uses optimal threshold function opt (Bi) ,to be used as a threshold for bidder i, which on set of bids B\bi returns the fixed price at which items should be sold to achieve revenue F.

• This action has a worst case input that causes it not to be competitive.

Deterministic Optimal Threshold Auction (cont.)

• An example for worst case input :– r bids with value h.– (h-1)r-1 bids with value 1.– Optimal single price auction take r bid with price h s.t F=

rh => threshold is h. (the second best threshold is 1 taking all bid for revenue of hr-1).

– Deterministic optimal threshold auction takes r high bids with price 1 s.t R=r. (the high are taken in price 1 because removing one h bids causes the opt. threshold to switch to 1)

– So R/F=1/h.

Upper bound for deterministic Bid-independent action

• Theorem: For any truthful deterministic bid-independent auction and any constant α there exist an input for which R/F =O(1/h) and αh F.

Truthful deterministic Auction are Bid-independent

• Lemma: Any truthful deterministic auction is bid independent

• Proof: – Bi = B\{i}– Bix = bi replaced with value x.– A(B) = the result of running auction A on set of

bids B.– Ai(B) – the result for bidder i( if i is rejected).– We will define function g as: g(x) = Ai(Bix).

Proof(cont.)

• It is left to show that g is except for some region (v, ), [v, ) or {} where it has the value v. This would imply A is bid-independent.

• If g(x)= for all x than we have empty interval {}.

• Otherwise, let b = inf{x:g(x) } and v=minx g(x). It is left to show b=v and for all b’>b, g(b’)=v

Proof(cont.)

1. v=b: v b since g(x) is defined as (loss) or g(x) x (a bid will win at price necessarily at most x). v b since else there might exist b’ s.t v<b’<b. The bidder with utility value b’ would be better off biding greater than b winning the auction and pay v which is less that b’. This is a contradiction of A been truthful. Thus, v=b.

2. For all b’>b, g(b’)=v: since that if g(b’) will be bigger than v, a bidder with utility b’ will be better off bidding b. This is a contradiction of A been truthful.

Upper bound for deterministic Bid-independent action

• Theorem: For any truthful deterministic bid-independent auction and any constant α there exist an input for which R/F =O(1/h) and αh F.

Why Simulation is needed?

• In terms of asymptotic worst-case performance, deterministic auctions are significantly worse that randomized auction => this is not to say deterministic auctions are bad to use for all input families.

• Constant factors (not necessarily tight) obtained in the analysis do not enable determination which auction does better.

• Theoretical analysis of the presented auctions for specific distributions is non-trivial.

Some simulation results

• On uniform and normal distribution (“average case” families) for large n the dual-price sampling optimal threshold and the deterministic optimal threshold have the best results.

• The above families have the property that any uniformly chosen random subset of the bids has the same distribution as the original. Because of this random sampling auction perform very good on such families.

• Weighted pairing is the worst auction for average case families.

Some simulation results (cont.)

• Even on contrived worst-case families the auction revenue is a large constant fraction of F.

• Sampling about a square root of the number of bids for the sampling auction seems to balance out the loss due to rejecting all of the sample and the loss due to having a non representative sample.

Extension for Bounded Supply

• The bounded supply case is a generalization of the unlimited supply case as items are available in unlimited supply when the number of available items is the same as the number of bidders.

• Denote the number of items available by k.• Denote by Fk the revenue in the bounded supply

case.• Let optk be the function that given a set of bids

return the optimal threshold that sells k item or less.

Extension for Bounded Supply(cont.)

• The single price sampling optimal threshold will be modified to use threshold function optmk/(n-m) on sample of size m. If more than k bids are satisfied, we arbitrary reject bids until there are only k left.

• The dual-price auction with sample size of m=n/2 use optk/2 so that about k/2 bids are selected from each of the sample and the non-sample.

• Both auction can be shown to be competitive.

Conclusion

• There exist truthful auction for unlimited supply market.

• Randomized auction are competitive in that they yield revenue that is within a constant factor of optimal fixed pricing.

• No deterministic auction is competitive in the worst case.

• The presented auctions compare favorably to fixed pricing with market analysis (simulation result)

Appendix: Competitive Auction for Multiple Digital Goods.

Problem Definition

• The problem of selling several items , with each item available in unlimited supply. Each bidder wants only a single item. Example: concurrent broadcast of several movies.

• The input for an auction:– Number of bidders n.– Number of items m.– Set of bids {aij}.

Problem Definition (cont.)

• Auction outcome: an assignment of a subset of wining bidders to items. Each bidder i in the subset is assigned a single item j and a sales price of at most aij.

• Stable auction: bidding Uij is a dominant strategy for bidder i. It maximize the bidder profit at the bidder’s utility value.

Note: the set of bidders strategies where bidder i bids uij for item j is a Nash equilibrium.

“Fixed Price” Auction

• Bidders supply bids aij, seller supplies sale prices rj 1 j m.

• Definition cij=aij-rj.• The auction assigns each bidder i to the item j with

the maximum cij. The sale price is rj.

• Lemma: Suppose the sale prices are set independently of input bids. Then fixed price auction is stable.

Optimal Sale Prices

• The problem of finding for a given set of bids and a set of prices an assignment such that the fixed price auction brings the highest revenue can be view as a matching problem.

• Translating this to an integer program we get the following linear program:max j iXij(aij-rj) s.t.

jXij 1 1 i n Xij 0 1 i n , 1 j m

Xij is one exactly when bidder i get item j.

Optimal Sale Prices (cont.)

• The dual problem is expressed it terms of pi, the profit of the corresponding bidders:min ipi s.t.

pi aij –rj 1 i n , 1 j mpi 0 1 i n

• If we combine both of these optimization problems we get:

Optimal Sale Prices(cont.)

Find Xij and Pi s.t

jXij 1 1 i n

Xij 0 1 i n , 1 j m

pi + rj aij 1 i n , 1 j m

ipi = j i Xij (aij - rj)

Optimal Sale Prices(cont.)

• Treating rj as variable the optimal sale price problem as the following mathematical programming problem:max j i Xij rj s.t.

rm=0jXij 1 1 i nXij 0 1 i n , m j mpi + rj aij 1 i n , m j mipi = j i Xij (aij - rj)

Extension of the random sampling auction

• Pick a random sample S of the set of bidders. Let N be the set of bidders not in the sample.

• Compute the optimal sale prices for S.• The result of the random sampling auction

is then just the result of running the fixed price auction on N using the sale prices computed in the previous step.

Extension of the Deterministic Auction

• Delete i from B.

• Compute optimal prices for the remaining bidders

• Choose the most profitable item for i under these prices.

• This is done independently for each bidder.

Conclusion

• The random sample auction is competitive.

• In order to studies this auction using simulation there is a need for a fast algorithm for calculation the optimal fixed pricing problem.