C OMPETITIVE A UCTIONS 1. W HAT WILL WE SEE TODAY ? Were the Auctioneer! Random algorithms Worst case analysis Competitiveness 2.

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COMPETITIVE AUCTIONS

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WHAT WILL WE SEE TODAY?

Were the Auctioneer! Random algorithms

Worst case analysis Competitiveness

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OUR PLAYGROUND

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

Digital goods

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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

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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

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BID-INDEPENDENT AUCTIONS

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BID-INDEPENDENT AUCTIONS

Intuition Masked vector

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

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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 b

t b

x t

p

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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

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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

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TRUTHFUL -> BID-INDEPENDENT

Theorem : A deterministic auction is truthful if and only if it is equivalent to a deterministic bid-independent auction.

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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

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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 truthfulness

p2

u2

u1

p1

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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

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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!

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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

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COMPETITIVENESSDOT

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ROLE MODELS

The competitive notion

Single Price Optimum:

Multi-price Optimum:

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DOT

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

DOT:

Calculates maximum for rest of the group

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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)

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WHERE DOT FAILS

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

100

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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

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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

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COMPETITIVENESS 2 and FmF

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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

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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) ]

2h f h

(1 2 )2

Hh

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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

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Similarly we define the sale of at least m items

( ) maxmm k n kF b kv

mF

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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

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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 b

n

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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

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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

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DETERMINISM SUCKS: PROOF

For ( )

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 n

m

n

F b

mm k F b

n

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CONCLUSION

Why worst case? Not truly random source

How competitive? F is too good

Why random? Because determinism is not good enough

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RANDOM AUCTIONS

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RANDOM AUCTIONS

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

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DSOT

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DSOT

Observation: truthful

C competitive to F(2) (without proof)

Unknown C, at least 4

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ECCENTRIC MILLIONAIRES EXAMPLE

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

b’ b’’

1M

1M+1

1M1

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ECCENTRIC MILLIONAIRES EXAMPLE

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

b’ b’’

1M

1M+1

1M1M+1

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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

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BETTER BOUNDS: SPECIAL CASE

Special case where b is bounded-range:

Then

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PROOF

Denote best sale price for at least r items

The price for

Than lets define

( )rF

( )rDSOT

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( )rDSOT

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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

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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

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SCS

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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’

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SCS COMPETITIVE

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

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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

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BOUNDED SUPPLY

If we only have k goods

Than we use k best bidders and run unlimited supply case

Competitive vs

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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)

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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

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COMPETITIVENESS IIis F the best benchmark?

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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?

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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

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HARD CODED AUCTIONS

Let b* be out bid specific bid

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

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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

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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

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QUESTIONS?