1 Deterministic Auctions and (In)Competitiveness Proof sketch: Show that for any 1  m  n there exists a bid vector b such that Theorem: Let A f be any symmetric deterministic auction defined by the bid-independent function f. Then A f is not competitive.

2 Deterministic Auctions and (In)Competitiveness (Cont.) Proof sketch (cont.): Fix m and n Consider bid vectors whose bids are all n and 1 Show that there is a bid vector b with k+1 ns such that Or else the solution is trivial. Thus:

3 Deterministic Auctions and (In)Competitiveness (Cont.) If k+1<m we have: and: Else - k+1  m and: resulting with which proves the theorem Proof sketch (cont.): Now:

4 Competitive Auctions via Random Sampling Randomly partition the bid vector b into two sets b ’ and b ’’ Compute p ’ based on b ’ and p ’’ based on b ’’ Assign the price p ’ for b ’’ and p ’’ for b ’ Two algorithms: DSOT SCS

5 Dual-Price Sampling Optimal Threshold Auction (DSOT) Uses as the price setting mechanism Constant competitive against F (2) The bound is weak for the general case Significantly better performance for some interesting special cases

6 DSOT – the Algorithm Input: bid vector b Output: Allocation vector x, price vector p Randomly partition b into b’ and b’’ Compute p’=opt(b’) and p’’ = opt(b’’) Use p’ as a threshold for b’’ Use p’’ as a threshold for b’

7 DSOT - Example b = opt(b’) = 7opt(b’’) = x’ = 0100 p’ = 110 x’’ = 770 p’’ = b’ = b’’ =

8 DSOT – Performance Analysis In the general case – DSOT is constant competitive against F (2) ; this bound is weak For some interesting special cases DSOT’s performance is much better Example: If b is bounded-range bid vector (b i  [1, h]) then

9 Sampling Cost-Sharing Auction (SCS) Uses CostShare C for setting the price At least 4-competitive against F (2) Definition (CostShare C ): Given a cost C and bids b, find the largest k such that the highest k bid’s value  C/k. Charge each C/k. CostShare C is truthful If (C  F (b) then CostShare C has revenue of C; Otherwise it has no profit

10 SCS – the Algorithm Input: bid vector b Output:Allocation vector x, price vector p Randomly partition b into b’ and b’’ Compute F ’= F (b’) and F ’’= F (b’’) Compute the auction results by running CostShare F ’ (b’’) and CostShare F ’ ’ (b’)

11 SCS - Example b = F (b’)=21 F (b’’)= x’ = p’ = 000 x’’ = 000 p’’ = b’ = b’’ =

12 SCS – Performance Analysis Proof: Assume F (b)=k·p then F ’ (b’)=k’·p’  k’·p and F ’’ (b’’)=k’’·p’’  k’’·p If F ’= F ’’ then F ’+ F ’’ F (b) and we are done Otherwise Theorem: SCS is 4-competitive and this bound is tight

13 SCS – Performance Analysis Proof (continue): Expected value of min(k’, k’’): Thus, the competitive ratio achieves its minimum of ¼ at k=2,3. as k increases, the ratio approaches ½

14 Bounded Supply We may sell no more than k items We wish to be competitive against F (m,k) : Reduction to the unlimited supply case: Reject any bid that is not among the k highest bids Run the unlimited supply auction on the rest

15 Another look at Competitive Analysis Thus far we have compared performance to F, the optimal single-price auction Is it “fair” to compare a dual-price auction to the optimal single-price auction? Theorem: for any monotone (truthful) randomized auction A, and for all bid vectors b, R A (b)=  i p i satisfies E[R]  F (b)

16 Intuition: Since b i  b j then b -i looks like a higher set of bids than b -j We would expect a higher set of bids to yield a higher price Monotonicity Definition (monotone auction): An auction is monotone if for any pair of bidders i and j with b i  b j and for any t  b i, we have Pr[(x i =1)  (p i  t)]  Pr[(x j =1)  (p j  t)]

17 Example: Consider the following auction A given by the bid-independent function f : Where’s the catch? Hard-Coded Auctions For any bid vector b, there exists a truthful auction A that satisfies A (b)= T (b)

18 Thus F is the optimal monotone function Monotonicity (Cont.) Theorem: Let A be any monotone truthful randomized auction. For all bid vectors, the revenue of A satisfies E[R]  F (b) DSOT, SCS, Vickrey auctions are all monotone; so is F

19 The notion of competitive auction was introduced Justification for using F was given It was shown that no deterministic auction may can be competitive 2 novel randomized auctions for the unbounded supply scenario, DSOT and SCS were introduced Reduction to the bounded supply was shown Summary

20 Cancelable auctions Envy-free auctions Almost truthful auctions Online auctions Related Work

21 Thank you!