Approximations and Truthfulness: The Case of MultiUnit Auctions

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Approximations and TruthfulnessThe Case of
Multi-Unit Auctions

• Shahar Dobzinski
• Based on a joint work with Noam Nisan (Hebrew U)
  and a work in progress with Shaddin Dughmi
  (Stanford)

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Algorithmic Mechanism Design

• Design efficient algorithms for selfish players.
• E.g., sponsored search, FCC spectrum auctions,
  sharing network resources, eBay auctions,
• Truthfulness a player is never better off by
  misreporting his true value.
• Cooperation is the best.
• profit (value for 10 TeraFlops) (price for 10
  TeraFlops)
• Weaker solution concepts later in this talk.
• This talk AMD via multi-unit auctions.
• Mainly, computational efficiency vs. incentives.

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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.
• Valuations are given as black boxes.
• 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
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Approximations and VCG

• Computer Science
• NP-complete to solve, but a (1e)-approximation
  exists.

• Game Theory
• The VCG Mechanism is a truthful mechanism for
  multi-unit auctions.

Can we obtain an algorithm that is both efficient
and truthful?
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Related Work

• Mualem-Nisan, Kothari-Parkes-Suri, Lavi-Swamy,
  Briest-Krysta-Vocking, Lavi-Mualem-Nisan,
  Balcan-Blum-Mansour,
• Mostly considered special cases.

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Our Results

• A deterministic truthful 2-approximation
  mechanism for multi-unit auctions.
• And this is the best that can be achieved (using
  our techniques)
• A (1e)-approximation algorithm that is truthful
  in expectation.

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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 profit of a bidder is the welfare of the
  allocation e.g., Bidder 1s profit is
  v1(o1)Sjgt1vj(oj) Sjvj(oj) OPT
• Using an approximation algorithm with VCG
  payments does not result in a truthful algorithm,
  unless the algorithm is maximal-in-range.
• MIR limit the range and fully optimize over the
  restricted range.

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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.

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The Approximation Guarantee

• Consider 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.

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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 at least
  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 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.

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A Lower Bound of 2 for Maximal in Range Algorithms

• Claim Let A be an MIR (2-e)-approximation
  algorithm for multi-unit auction with 2 bidders.
  Then, As range contains all allocations.
• Proof
• Otherwise, there is an allocation (k,m-k) that is
  not in As 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.

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A General Lower Bound?

• Thm charecterization if there are only 2
  bidders and all items are allocated, then every
  truthful mechanism must be maximal in range.
  Lavi-Mualem-Nisan, Dobzinski-Sundararajan.
• Thm optimization a maximal in range algorithm
  cannot provide an approximation ratio better than
  2 in poly time.
• Major Open Question in Algorithmic Mechanism
  Design is there a truthful poly-time mechanism
  with an approx ratio better than 2?

Characterize
Optimize
Lower Bound
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Whats Next?

• How can we improve the approximation ratio?
• Relax the notion of truthfulness.
• Undominanted strategies, differential privacy,
• Our relaxed notion 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.
• Theorem There exists a (1e)-approximation
  algorithm that is truthful in expectation.

Reminder in this talk, poly time poly of
value queries.
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Truthfulness in Expectation in Multi-Unit Auctions

• Maximal in range in expectation
• The range is a set of distributions over
  allocations.
• The algorithm always chooses 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 the algorithm is
  truthful in expectation.

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The Range

• Only weighted allocations choose the pure
  allocation s(s1, , sn) with probability ws,
  with probability 1-ws allocate nothing.
• The Range (bonus simplicity)
• The allocation (s1, , sn) has weight
  wt(1-e)(1d)t, if all sis are multiples of 2t,
  for t 0.
• Only log(m) weights, d log(1/1-e) / log m.
• Approximation ratio every pure allocation is in
  the range and has a weight of at least 1-e.
• Truthfulness by optimizing over the range we get
  an MIR algorithm.

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The Algorithm Some Assumptions

• Lemma WOPT can be found efficiently.
• WLOG, the weight of the optimal (weighted)
  allocation is wi.
• Only log(m) weights, so we can try all possible
  weights.
• This talk assume WOPT has weight w0.
• At least one bidder receives an odd number of
  items in WOPT.

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The Algorithm
Some valuation function

• Each bidder transmits a step function based on
  the valuation.
• Round down each value to the nearest power of
  (1d/2).
• Each bidder transmits a poly of values.
• Each bidder transmits n places after each step.
• I.e., if there is a green point at v(5),
  transmit v(5) to v(5n)

Value
Items
The algorithm chooses the best weighted
allocation that consists only of bundles with
transmitted values.
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Correctness

• Let WOPT(o1,,on).
• Def A step allocation is an allocation (s1,,sn)
  such that each vi(si) is a beginning of a step.
• Lemma It is possible to remove less than n items
  from WOPT and obtain a step allocation (s1,,sn).
• Proof Suppose not.
• Round down each oi to the nearest step oi. We
  removed at least n items.
• Add one item to each oi that is odd ai.
• w1Svi(ai) (1d)w0 Svi(ai) (1d)w0Svi(oi)
  (1d)WOPT/(1d/2) gt WOPT

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Summary

• We presented a truthful deterministic
  2-approximation mechanism, and proved a matching
  lower bound for MIR algorithms.
• We discussed a possible way of proving lower
  bounds for all truthful algorithms.
• Characterize, then optimize.
• We relaxed the notion of truthfulness, and
  presented a (1e)-approximation algorithm that is
  truthful in expectation.

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Open Problems

• Prove lower bounds for deterministic mechanisms,
  not just MIR ones.
• Might require us to obtain better
  characterization results.
• Are there any good truthful in expectation
  mechanisms for other settings?
• A constant approximation mechanism for
  combinatorial auctions with submodular bidders?

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My Research

• The power of polynomial time truthful mechanisms
• What is the computational burden of truthfulness?
• Approximation Algorithms for combinatorial
  auctions
• Sponsored search auctions
• Budgets and online auctions
• Cost Sharing mechanisms
• How well can we share the cost of a service?
• Congestion games and networks
• Sharing network resources and incentives.
• Voting
• How frequently can elections be manipulated?

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Dominant-Strategy Truthfulness

• Mechanism Allocation Rule (s1, ,
  sn)f(v1,,vn) and player payments pi(v1 vn)??.
• Truthfulness a rational player will always
  report his true valuation to the mechanism.
• For every profile of valuations, you do not gain
  by lying
• ? i ? v1 vn ? vi vi(si)-p vi(si)-p
• where (s1, , sn) f(vi v-i), p pi(vi v-i),
  (s1, , sn) f(vi v-i), ppi(vi v-i).
• Weaker notions exist. We will discuss them later.