Approximations and Truthfulness: The Case of Multi-Unit Auctions

1. Approximations and Truthfulness:The Case of Multi-Unit Auctions ShaharDobzinski Joint work with Noam Nisan and with ShaddinDughmi

2. Auctions

3. Clean Air Auction • 1990’s: The US government decided to decrease the atmospheric levels of sulfur dioxide. • From 18.9 million tons to 9 million tons. • Power plants tried to avoid legislation. • Estimated cost of a decrease in a single ton: 700$ • Solution: (annual) auction of emission allowances. • Cost of a 1-ton allowance: ~70$.

4. Definition of Multi-Unit Auctions • n bidders, m (homogenous) items. • vi(s) denotes the value of bidder i for a bundle of s items (represented by black boxes). • Normalization: vi(0) = 0 • Monotonicity: vi(s+1) ≥ vi(s) Goal: find an allocation (s1,…,sn), Ssi≤m, that maximizes the welfareSivi(si). Fast running time: running time must be polynomial in n and log m. A generalization of Knapsack.Input: n objects, each has size si and value vi , capacity m. Goal: Find a maximum-value subset of the objects with total size of at most m.

5. Computation: NP-hard to solve, but a (1-e)-approximation algorithm exists. • Incentive Compatibility: • Truthfulness: a player is never better off by misreporting his true value. For example, second price auction. • Technically: dominant strategies, private values, quasi-linear utilities. Incentives: The VCG mechanism is a truthful mechanism for multi-unit auctions. Can we obtain an algorithm that is fast, truthful and approximates the welfare well?

6. Related Work Mu’alem-Nisan, Kothari-Parkes-Suri, Lavi-Swamy, Briest-Krysta-Vocking, Lavi-Mu’alem-Nisan, Balcan-Blum-Mansour, …

7. Theorem: There exists a poly time deterministic truthful ½-approximation mechanism. Some evidence that deterministic truthful mechanisms cannot achieve a better approximation ratio in polynomial time. Theorem: There exists a truthful poly time randomized (1-e)-approximation mechanism.

8. A truthful mechanism (VCG): • Find the optimal allocation (o1,…,on). Assign the bidders items accordingly. • Pay each bidder i: Sj≠ivj(oj). Proof of truthfulness: the profit of a bidder is the welfare of the allocation. E.g., bidder 1’s profit is v1(o1)+Sj>1vj(oj) = Sjvj(oj) = OPT • Approximation algorithm + VCG payments? • Truthful only if the algorithm is maximal-in-range. Maximal in Range: limit the range and fully optimize over the restricted range.

9. The mechanism: • Split the items into n2equi-sizedbundles each of size m/n2. • Allocate these bundles optimally. Lemma: the algorithm is truthful. Using VCG payments, since the algorithm is maximal-in-range. Lemma: the algorithm runs in polynomial time. By using dynamic programming. Lemma: the algorithm guarantees an approximation ratio of ½.

10. Lemma: the algorithm guarantees an approximation ratio of ½. • WLOG, all items are allocated in the optimal solution (o1,…on). Let o1 ≥ m/n. • Claim: There exists an allocation in the range that holds at least half of the value of the optimal solution. • Proof case 1: If v1(o1) > Si>1vi(oi), allocate all items to bidder 1 and get a ½ approximation. • Proof case 2: If v1(o1) ≤ Si>1vi(oi), round up each oi to the nearest multiple of m/n2, and set o1 to 0. We get a ½ approx.We added at most (m/n2)*n=m/n items, but removed at least m/n items, thus the allocation is valid and in the range.

11. Theorem: Every maximal-in-range (½+e)-approx algorithm requires at least mqueries to the black boxes. The proof follows from the following two claims: Claim: Let A be an MIR (½+e)-approximation algorithm for multi-unit auctions with 2 bidders. Then, A’s range must contain all allocations. Claim [Nisan-Segal]: Optimally solving multi-unit auctions (even with only 2 bidders) requires at least m queries to the black boxes.

12. Claim: Let A be an MIR (½+e)-approximation algorithm for multi-unit auctions with 2 bidders. Then, A’s range must contain all allocations. Proof: Otherwise, there is an allocation (k,m-k) that is not in A’s range. Consider the following instance: Bidder 1 values a bundle of at least k items with 1 (and 0 o/w), and Bidder 2 values a bundle of at least m-k items with 1 (and 0 o/w). The optimal welfare is 2, but A provides welfare of at most 1.

13. Proving Impossibilities:Characterize and Optimize Step 1 (characterize): (essentially)every truthful mechanism with an approximation ratio better than ½ is maximal in range. Step 2 (optimize): a maximal in range algorithm cannot provide an approximation ratio better than ½ in poly time Characterize + Optimize = Impossibility

14. What’s Next? • Truthfulness in Expectation: Bidding truthfully maximizes the expected profit of each bidder. Expectation is over the internal random coins of the algorithm. • Good for risk-neutral bidders. • Stronger than Bayesian incentive compatibility. Theorem: there exists a (1-e)-approximation mechanism that is truthful in expectation.

15. Maximal In Distribution Range: The range is a set of distributions over allocations. Choose the distribution in the range that maximizes the expected social welfare Randomly select an allocation according to the distribution. • Pay each bidder the sum of the values of the others in the realized allocation. • The expected profit of a bidder is the welfare of the best distribution, hence truthfulness in expectation.

16. The range contains only weighted allocations. Each allocation s=(s1,…,sn) will have a weight ws. With probability ws allocate according to s and with probability 1-ws allocate nothing. The range (“reward simplicity”): Weights have the form wt=(1-e)(1+d)t, where d= log(1/1-e) / log m The weight of the allocation s=(s1,…,sn) is wt, where t is the maximal integer s.t. all si’s are multiples of 2t. Relaxation: the optimal distribution has weight w0. At least one bidder is allocated an odd number of items. We have good approx ratio and truthfulness. Poly time?

17. The Algorithm • Each bidder computes a step function based on the valuation: round down each value to the nearest power of (1+d/2).Poly # of “breakpoints”. • Each bidder transmits each breakpoint and n subsequent values (“neighbors”).I.e., if there is a breakpoint at v(5), transmit v(5),v(6),…,v(5+n) Some valuation function Value Items Select the best weighted allocation that consists of only breakpoints and neighbors.

18. Correctness Lemma: Let WOPT= w0*(o1,…,on). Each oi is either a breakpoint or a neighbor of such. Proof: Suppose not. Round down each oi to the nearest breakpoint o’i. We removed at least n items. Add one item to each o’i that is odd: ai. w1Svi(ai) = (1+d)w0Svi(ai) ≥ (1+d)w0Svi(o’i) ≥ (1+d)WOPT/(1+d/2) > WOPT

19. Summary A deterministic truthful ½ approximation mechanism. An impossibility for MIR mechanisms. hardness for almost all truthful mechanisms. Characterize and Optimize. A truthful in expectation randomized (1-e)-approximation mechanism.

20. Open Questions Are there any good MIDR mechanisms for other settings? An O(1)-approximation mechanism for combinatorial auctions with submodular bidders? See [Dughmi-Fu-Kleinberg] for initial results.

21. Basic Requirements from Auctions Social goal implementation: maximize welfare, revenue, fairness, … Fast running time: polynomial. • Incentive Compatibility: • Truthfulness: a player is never better off by misreporting his true value. For example, second price auction. Technically: dominant strategies, private values, quasi-linear utilities. This talk: auction design via multi-unit auctions.

22. Modern Auctions I: eBay Many simultaneous auctions, many buyers and sellers.

23. Modern Auctions II: Sponsored Search Many ads and keywords, online, automatic bidding, budgets, …