Mechanisms for MultiUnit Auctions

1
Mechanisms for Multi-Unit Auctions

• Shahar Dobzinski
• Joint work with Noam Nisan

2
Multi-Unit Auctions

• n bidders, m (identical) items
• For each bidder i, vi(s) denotes the value of
  bidder i for getting a bundle of s items.
• Normalization vi(0) 0
• Monotonicity vi(s1) vi(s)
• Goal find an allocation of the items (s1,,sn),
  Ssim, that maximizes the welfare Sivi(si)
• Algorithms are required to run in time polynomial
  in n and log m.
• Special case knapsack.

Input n objects, each one with size si and value
vi , capacity m. Goal Find a maximum-value
subset of the objects with total size of at most m
3
Previous Results

• NP-Hard (a reduction from knapsack), but an FPTAS
  exists.
• For single-minded bidders there is a truthful
  FPTAS Briest-Krysta-Vocking.
• A single-minded bidder values si items or more
  with a value of vi, and every other bundle with
  0.
• FPTAS A (1e)-approximation in time polynomial
  in n, log m, and 1/e.
• Other results Kothari-Parkes-Suri, Lavi-Swamy,
  Mualem-Nisan, Lavi-Mualem-Nisan,

4
Our Results

• A truthful 2-approximation mechanism for general
  bidders.
• Valuations are given as black boxes.
• And this is the best that can be achieved (using
  our techniques)
• A multi-parameter problem!
• A truthful PTAS for k-minded bidders.
• A (1e)-approximation In time polynomial in k,
  log m, and n.
• And this is the best that can be achieved (using
  our techniques)

5
VCG (applied to multi-unit auctions)

• A truthful mechanism for multi-unit auctions
  (VCG)
• Find the optimal allocation (o1,,on). Assign the
  bidders items accordingly.
• Pay each bidder i Sj?ivj(oj).
• Proof (of truthfulness)
• The utility of a bidder is the welfare of the
  allocation e.g., Bidder 1s utility is
  v1(o1)Sjgt1vj(oj) Sjvj(oj) OPT
• VCG is truthful iff the algorithm is
  maximal-in-range Nisan-Ronen
• MIR limit the range and fully optimize over the
  restricted range.

6
A 2-Approximation Truthful Algorithm for
Multi-Unit Auctions

• The mechanism split the items into n2 equi-sized
  bundles each of size m/n2. Optimally allocate
  these bundles.
• Truthfulness follows since the algorithm is
  maximal in its range (using VCG prices).
• Lemma The algorithm runs in poly time.
• Follows because this is an instance where the
  number of items is polynomial.
• Lemma The algorithm provides an approximation
  ratio of 2.

7
The Approximation Guarantee

• Look at the optimal solution (o1,on).
• WLOG, all items are allocated.
• Suppose o1 is the bundle with the largest of
  items. Clearly, o1 m/n.
• There exists an allocation in the range that
  holds at least half of the value of the optimal
  solution. Two Possible Cases
• If v1(o1) gt Sigt1vi(oi), then by allocating all
  items to bidder 1 we get a 2 approximation.
• If v1(o1) Sigt1vi(oi), round up each oi to the
  nearest multiple of m/n2, and set o1 to 0. We get
  a 2 approx.
• We added at most (m/n2)nm/n items, but removed
  at least m/n items ? legal and in the range.

8
Summary

• A 2-approximation algorithm for multi-unit
  auctions with general bidders.
• PTAS for most interesting special cases.
• Best results possible using maximal-in-range
  mechanisms.
• Main open question prove lower bounds on general
  mechanisms, not just MIR mechanisms.

9
A Lower Bound of 2 for Maximal in Range Algorithms

• Theorem Every maximal-in-range
  (2-e)-approximation algorithm for multi-unit
  auctions (with general bidders) requires m
  queries to the black boxes.
• Proof follows from the two claims
• Claim Let A be an MIR (2-e)-approximation
  algorithm for multi-unit auctions with 2 bidders.
  Then, As range must be full.
• Claim (Nisan-Segal) Optimally solving multi-unit
  auctions (even with only 2 bidders) requires m
  queries to the black boxes.

10
A Lower Bound of 2 for Maximal in Range Algorithms

• Claim Let A be a MIR (2-e)-approximation
  algorithm for multi-unit auction with 2 bidders.
  Then, As range must be full.
• Proof
• If As range is not full then there exists an
  allocation (k,m-k) that is not in As range.
• Define 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, while the welfare A
  provides is at most 1.