1

COMPETITIVE AUCTIONS

2

WHAT WILL WE SEE TODAY? Were the Auctioneer! Random algorithms

Worst case analysis Competitiveness

3

OUR PLAYGROUND Unlimited number of indivisible goods No value for the auctioneer Truthful auctions

Digital goods

4

BEFORE WE BEGIN Normal Auctions (single round sealed bid)

utility vector u bid vector b payment vector p Auction A

Profit is sum of payments

5

RANDOM TRUTHFULNESS Reminder: Truthful auctions are auctions

where each bidder maximizes his profit when bids his utility

Random is probability distribution over deterministic auctions

Random Strong Truthfulness One natural approach Our chosen approach A randomized auction is truthful if it can be

described as a probability distribution over deterministic truthful auctions

6

BID-INDEPENDENT AUCTIONS

7

BID-INDEPENDENT AUCTIONS Intuition Masked vector

f a function from masked vectors to prices Every buyer is offered to pay

8

AUCTION Auction 1: Bid-independent Auction: Af(b)

1. ( )2.if then 2.1. 1 and p 2.1 else x 0 and 0

i i

i i

i i i

i i

t f bt bx t

p

9

EXAMPLES Bid vector for buying Lonely-Island new song 4 bets

What have we got? 1-item vickery

For k’th largest bid we get K- item vickery

( ) max( )i if b b

10

BID INDEPENDENT -> TRUTHFUL

We are offered T(=20) what should we bid? If U(=15) < T we cant win If U(=30) >= T any bid >= T will win Either way U maximizes bidder’s profit

T U

max profit

11

TRUTHFUL -> BID-INDEPENDENT Theorem : A deterministic auction is truthful

if and only if it is equivalent to a deterministic bid-independent auction.

12

TRUTHFUL->BID-INDEPENDENT For bid vector b and bidder i we fix all bids

except bi

Lemma1 For each x where i wins he pays same p

Lemma2i wins for x>p (possibly for p)

1 1 1( ,..., , , ,..., )xi i i nb b b x b b

13

LEMMA 1 PROOF Lemma1: i pays p Assume to the contrary

x1,x2 where i pays p1>p2 Than if Ui = x1 i should lie and tell x2

=>In contrast to A’s truthfulnessp2

u2

u1

p1

14

LEMMA2:PROOF Lemma2: for each x>p (and possibly p) x

wins Assume to the contrary w exists

w>p w wins

x exists such that x>p x doesn’t win

if U=x i should lie and say w => In contrast to A’s truthfulness

P wx

15

TRUTHFUL->BID-INDEPENDENT Define

Than for any bid b

For bid b if i in A wins and pays p than also in Af If loses than

p doesn’t exist or bi < p

;if i can win for any x( ) {

;elsei

pf b

fA A

Bid Indepndent is truthful!

16

LETS SHAKE THINGS UP Reminder:

Random Auctions Random Truthful Auctions

A randomized bid-independent auction is a probability distribution over bid-independent auctions

=> A randomized auction is truthful iff it is equivalent to a randomized bid-independent auction

17

COMPETITIVENESSDOT

18

ROLE MODELS The competitive notion

Single Price Optimum:

Multi-price Optimum:

19

DOT Deterministic Optimal Threshold single-priced Define opt(b) as the optimum single price

DOT:

Calculates maximum for rest of the group

20

WHERE DOT IS OPTIMAL

Bids range from [0$,50$] Bids are i.i.d

DOT optimal for a wide range of problems! For any bounded support i.i.d(without proof)

21

WHERE DOT FAILS

n bidders(100 bidders) n/a bid a>>1(1 high paying bidder) Else bids 1

100

22

WHERE DOT FAILS For each a bidder :

(n/a-1) a-bidders profit for p=a is n-a but for p=1 is n-1 p = 1

For each 1 bidder n/a a-bidders profit for p=1 is n-1 but for p=a is n p = a

Profit is n/a (number of a bidders)

100

23

DOT CONCLUSION Why are we talking worst case? DOT prevails in Bayesian model Loses in worst case When not safe to assume true random source

Competitive outlook is logical

24

COMPETITIVENESS 2 and FmF

25

F-COMPETITIVE FAILURE Lemma: For any truthful auction Af and any

β≥1, there is a bid vector b such that the expected profit of Af on b is less than F(b)/β

26

PROOF 2 bidders Define h the smallest value such that

Lets consider the bid {1,H} where H=4βh>1 Profit is at most

For H bidder : For 1 bidder : 1

( ,1); 1b x x

11;Pr[ (1) ]2

h f h

(1 2 )2H h

27

Set our eyes lower 2-optimal single price bid The optimal bids that sells at least 2 items

Same as f(b) unless there is one bidder with Hugh utility

22( ) max k n kF b kv

2F

28

Similarly we define the sale of at least m

items

( ) maxmm k n kF b kv

mF

29

Β-COMPETITIVE Definition: We say that auction A is β-

competitive against F-m if for all bid vectors b, the expected profit of A on b satisfies

( )[ ( )]mF bE A b

30

DETERMINISM SUCKS Were going to show that no deterministic auction

is β competitive Theorem: Let Af be any symmetric deterministic

auction defined by bin-independent function f. Then Af is not competitive. For any m,n there exists a bid vector b of length n such the Af’s profit is at most

Symmetric auction: order of bids doesn’t matter For example, consider F(2). We can find a bid

vector at length 8 such that Af’s profit is at most F(2)/4

( ) ( )m mF bn

31

DETERMINISM SUCKS: PROOF Lets look at specific m,n at a specific auction

Af Consider bid b where all bids are n or 1

Let f(j) be the price where j bids are n n – 1 – j bid 1

for f(0) > 1 Consider the bids where all bids are 1

32

DETERMINISM SUCKS: PROOF k in 0..n-1 the largest integer where f(k) <= 1 We build a bid with

(k+1) n-bids (n – k – 1) 1-bids

1-bidders lose ( f(k+1) > 1) n-bidders win Profit : (k+1)f(k) < k + 1

33

DETERMINISM SUCKS: PROOFFor ( )

if k m-1 ( ) and condition holds

A (b) <k+1 m=F *

if k m then ( ) n*(k+1)

A (b) <k+1 *( 1) ( )*

m

m

mf

m

mf

F b

F b nmn

F bmm k F bn

34

CONCLUSION Why worst case?

Not truly random source How competitive?

F is too good Why random?

Because determinism is not good enough

35

RANDOM AUCTIONS

36

RANDOM AUCTIONS Split the bid vector b in two: b’, b’’ Use each part to build auction for the other

37

DSOT

38

DSOT Observation: truthful

C competitive to F(2) (without proof)

Unknown C, at least 4

39

ECCENTRIC MILLIONAIRES EXAMPLE Small-time bidders bid small (1) 2 Eccentric millionaires bid h,h+e

b’ b’’1M

1M+1

1M1

40

ECCENTRIC MILLIONAIRES EXAMPLE Small-time bidders bid small (1) 2 Eccentric millionaires bid h,h+e

b’ b’’1M

1M+1

1M1M+1

41

ECCENTRIC MILLIONAIRES EXAMPLE F(2) profit is 2h(= 2M) profit is h * Pr[2 high bids are split between

auctions] = h/2(=M/2)

Competitive Ratio of 4

42

BETTER BOUNDS: SPECIAL CASE Special case where b is bounded-range:

Then

43

PROOF Denote best sale price for at least r

items

The price for

Than lets define

( )rF

( )rDSOT

44

( )rDSOT

45

So, in special cases it has a very good bound In worst case, it is C-competitive C is worse than 4

( )rDSOT

2

2

: there is an absolute constant C, such that for any 0

is (1+ ) competitive again F , with probability at least 1-em C mm m

Theorem

DSOT

46

SCS Sampling Cost-sharing CostShare-C: if you have k bidders (highest)

which are willing to pay C collectively (bid>C/k). Charge each for C/k

CostShare is truthful For profit is C, else 0 I know exactly how much

I want to make, regardless of bids

47

SCS

48

SCS COMPETITIVE if F’=F’’ profit is at least F’F Auction profit is R = min(F’,F’’) Suppose F’<F’’

b’ cannot achieve F’’ b’’ profit is F’

49

SCS COMPETITIVE Suppose F(2) results is kp Uniform divison between b’ and b’’: k’ and k’’

50

COMPETITIVE RATIO Begins as ¼ Approaches ½

Tight proof Consider 2 high bids h,h+e

But we always throw half Can we improve? Yes, Costshare(rF’) and Costshare(rF’’) Competitive ratio is 4/r

51

BOUNDED SUPPLY If we only have k goods

Than we use k best bidders and run unlimited supply case

Competitive vs

52

BOUNDED-SUPPLY TRUTHFULNESS none of the bidders win at a price lower than

the highest ignored bid.

Use k-vickery to get p-v use auction of unlimited supply on winners get auction price p-A

use price max(pv,pA)

53

UP TILL NOW Bid independent is truthful Worst case outlook Our benchmarks: F,T Deterministic is just now good enough competitiveness against F(2)

Examples of random algorithms DOST: C-competitive SCS : 4-competitive

54

COMPETITIVENESS IIis F the best benchmark?

55

MULTI-PRICE F is best single price F(2) comparable to F

What about using T? T is only O(log(n)) better

Mabye other multi-priced?

56

MONOTONE FUNCTIONS F is better than all monotone auctions

Non-monotone example: Hard-coded actions Lets take b* such that half bid 1 and half bid h Lets create function which maximizes profit

Acts as omniscient on b* Poorly on other results Lets generalize

57

HARD CODED AUCTIONS Let b* be out bid specific bid

will maximize profit on b* bad profit on bids that differ in 1

58

MONOTONE FUNCTIONS

Basically, if you bid more you will pay less makes sense, for is higher for the lower bidder DOT,DSOT,SCS,

Vickery are monotone

ib

59

SUMMARY Bid independent is truthful Worst case outlook

competitiveness against F(2) use of random auctions

Examples of random algorithms DOST: C-competitive SCS : 4-competitive

F is a good benchmark

60

QUESTIONS?