C OMPETITIVE A UCTIONS 1

W HAT WILL WE SEE TODAY ? Were the Auctioneer! Random algorithms Worst case analysis Competitiveness 2

O UR PLAYGROUND Unlimited number of indivisible goods No value for the auctioneer Truthful auctions Digital goods 3

B EFORE WE BEGIN Normal Auctions (single round sealed bid) utility vector u bid vector b payment vector p Auction A Profit is sum of payments 4

R ANDOM T RUTHFULNESS 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 5

B ID - INDEPENDENT A UCTIONS 6

Intuition Masked vector f a function from masked vectors to prices Every buyer is offered to pay 7

A UCTION 8 Auction 1: Bid-independent Auction: Af(b)

E XAMPLES 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 9

B ID 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 10 TU max profit

T RUTHFUL -> B ID - INDEPENDENT Theorem : A deterministic auction is truthful if and only if it is equivalent to a deterministic bid- independent auction. 11

T RUTHFUL ->B ID - 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 Lemma2 i wins for x>p (possibly for p) 12

L EMMA 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 13 p2 u2 u1 p1

L EMMA 2:P ROOF 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 14 Pwx

T RUTHFUL ->B ID - 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 Bid Indepndent is truthful! 15

L ETS 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 16

C OMPETITIVENESS DOT 17

R OLE MODELS The competitive notion Single Price Optimum: Multi-price Optimum: 18

DOT Deterministic Optimal Threshold single-priced Define opt(b) as the optimum single price DOT: Calculates maximum for rest of the group 19

W HERE 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) 20

W HERE DOT FAILS n bidders(100 bidders) n/a bid a>>1(1 high paying bidder) Else bids

W HERE 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)

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 23

C OMPETITIVENESS 24

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 25

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 26

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 27

Similarly we define the sale of at least m items 28

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 29

D ETERMINISM 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 30

D ETERMINISM 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 31

D ETERMINISM 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

D ETERMINISM SUCKS : PROOF 33

C ONCLUSION Why worst case? Not truly random source How competitive? F is too good Why random? Because determinism is not good enough 34

R ANDOM A UCTIONS 35

R ANDOM A UCTIONS Split the bid vector b in two: b’, b’’ Use each part to build auction for the other 36

DSOT 37

DSOT Observation: truthful C competitive to F(2) (without proof) Unknown C, at least 4 38

E CCENTRIC MILLIONAIRES EXAMPLE Small-time bidders bid small (1) 2 Eccentric millionaires bid h,h+e b’b’’ 39 1M 1M+1 1M 1

E CCENTRIC MILLIONAIRES EXAMPLE Small-time bidders bid small (1) 2 Eccentric millionaires bid h,h+e b’b’’ 40 1M 1M+1 1M 1M+1

E CCENTRIC 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 41

B ETTER BOUNDS : SPECIAL CASE Special case where b is bounded-range: Then 42

P ROOF Denotebest sale price for at least r items The price for Than lets define 43

44

So, in special cases it has a very good bound In worst case, it is C-competitive C is worse than 4 45

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 46

SCS 47

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’ 48

SCS COMPETITIVE Suppose F(2) results is kp Uniform divison between b’ and b’’: k’ and k’’ 49

C OMPETITIVE R ATIO 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 50

B OUNDED SUPPLY If we only have k goods Than we use k best bidders and run unlimited supply case Competitive vs 51

B OUNDED - 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) 52

U P TILL N OW 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 53

C OMPETITIVENESS II is F the best benchmark? 54

M ULTI - 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? 55

M ONOTONE 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 56

H ARD CODED AUCTIONS Let b* be out bid specific bid will maximize profit on b* bad profit on bids that differ in 1 57

M ONOTONE F UNCTIONS Basically, if you bid more you will pay less makes sense, for is higher for the lower bidder DOT,DSOT,SCS, Vickery are monotone 58

S UMMARY 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 59

Q UESTIONS ? 60