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## Competitive Auctions

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**What will we see today?**• Were the Auctioneer! • Random algorithms • Worst case analysis • Competitiveness**Our playground**• Unlimited number of indivisible goods • No value for the auctioneer • Truthful auctions • Digital goods**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**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**Bid-independent Auctions**• Intuition • Masked vector • f a function from masked vectors to prices • Every buyer is offered to pay**Auction**• Auction 1: Bid-independent Auction: Af(b)**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**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 max profit T U**Truthful -> Bid-independent**• Theorem : A deterministic auction is truthful if and only if it is equivalent to a deterministic bid-independent auction.**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 • Lemma2 i wins for x>p (possibly for p)**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 p1 p2 u2 u1**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 x w**Truthful->Bid-independent**Bid Indepndent is truthful! • 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**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**Competitiveness**DOT**Role models**• The competitive notion • Single Price Optimum: • Multi-price Optimum:**DOT**• Deterministic Optimal Threshold • single-priced • Define opt(b) as the optimum single price • DOT: • Calculates maximum for rest of the group**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)**Where DOT fails**• n bidders(100 bidders) • n/a bid a>>1(1 high paying bidder) • Else bids 1 100**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**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**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**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**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**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**Determinism sucks**• Were going to show that no deterministic auction is βcompetitive • 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**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**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**Conclusion**• Why worst case? • Not truly random source • How competitive? • F is too good • Why random? • Because determinism is not good enough**Random Auctions**• Split the bid vector b in two: b’, b’’ • Use each part to build auction for the other**DSOT**• Observation: truthful • C competitive to F(2) (without proof) • Unknown C, at least 4**Eccentric millionaires example**• Small-time bidders bid small (1) • 2 Eccentric millionaires bid h,h+e b’b’’ 1M 1 1M 1M+1**Eccentric millionaires example**• Small-time bidders bid small (1) • 2 Eccentric millionaires bid h,h+e b’b’’ 1M+1 1M 1M 1M+1**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**Better bounds: special case**• Special case where • b is bounded-range: • Then**Proof**• Denote best sale price for at least r items • The price for • Than lets define**So, in special cases it has a very good bound**• In worst case, it is C-competitive • C is worse than 4**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**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’**SCS competitive**• Suppose F(2) results is kp • Uniform divison between b’ and b’’: k’ and k’’**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