Truthful Randomized Mechanisms for Combinatorial Auctions

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Truthful Randomized Mechanisms for Combinatorial
Auctions

• Speaker Michael Schapira
• Joint work with Shahar Dobzinski and Noam Nisan

Hebrew University
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Algorithmic Mechanism Design

• Algorithmic Mechanism Design deals with designing
  efficient mechanisms for decentralized
  computerized settings Nisan-Ronen.
• Takes into account both the strategic behavior of
  the different participants and the usual
  computational efficiency considerations.
• Target applications protocols for Internet
  environments.

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Combinatorial Auctions

• m items for sale.
• n bidders, each bidder i has a valuation function
  vi2M?R.
• Common assumptions
• Normalization vi(?)0
• Monotonicity S?T ? vi(T) vi(S)
• Goal find a partition S1,,Sn such that the
  total social-welfare Svi(Si) is maximized.

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Challenges

• Computer science compute an optimal allocation
  in polynomial time.
• Game-theory take into account that the bidders
  are strategic.

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Computer Science The Complexity of Combinatorial
Auctions

• For any constant e gt 0, obtaining an
  approximation ratio of min(n1-e, m½-e) is hard
• NP-hard even for simple valuations
  (single-minded bidders).
• Requires exponential communication (Nisan-Segal).
• Several O(m½)approximation algorithms are known.

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Game Theory Handling the Strategic Behavior of
the Bidders

• Our solution concept dominant strategy
  equilibrium.
• Due to the revelation principle we limit
  ourselves to truthful mechanisms.
• Implementable using VCG!
• Are we done?

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A Clash between Computer Science and Game Theory

• VCG requires finding the optimal allocation, but
  it is hard to calculate this allocation!
• Why not use an approximation algorithm for
  calculating (approximate) VCG prices?
• Unfortunately, incentive-compatibility is not
  preserved (Nisan-Ronen).
• We need other techniques!

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Deterministic Mechanisms

• We know how to design a truthful m½-approximation
  algorithm only for combinatorial auctions with
  single-minded bidders (Lehmann-Ocallaghan-Shoham)
  .
• This approximation ratio is tight.
• Only two results are known for the
  multi-parameter case
• A pair of VCG-based algorithms for the general
  case Holzman-Kfir Dahav-Monderer-Tennenholtz
  and for the complement-free case
  Dobzinski-Nisan-Schapira. Both are far from
  what is computationally possible.
• A non-VCG mechanism for auctions with many
  duplicates of each good Bartal-Gonen-Nisan.
• Theorem (wanted) There exists a polynomial time
  truthful O(m½)-approximation algorithm for
  combinatorial auctions.

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Randomness and Mechanism Design

• Randomness might help.
• Nisan Ronen show a randomized truthful
  7/4-approximation mechanism for the makespan
  problem with two players. They also show that any
  deterministic mechanism can not achieve an
  approximation ratio better than 2.

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On Randomized Mechanisms

• Two notions for the truthfulness of randomized
  mechanisms
• universal truthfulness a distribution over
  truthful deterministic mechanisms (stronger)
• Truthfulness in expectation truthful behavior
  maximizes the expected profit (weaker)
• Risk-averse bidders might benefit from untruthful
  behavior.
• The outcomes of the random coins must be kept
  secret.

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Previous Results and Our Contribution

• Lavi Swamy presented a randomized
  O(m½)-approximation mechanism that is truthful in
  expectation. We prove the following theorem
• Theorem There exists an O(m½)-approximation
  mechanism that is truthful in the universal
  sense.
• Actually, our result is stronger (details to
  follow).

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Our Mechanism An Overview

• We will describe our mechanism in several steps.
• First, assume that the value of the optimal
  solution, OPT, is known.

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Two Possible Cases

• Fix an optimal solution (OPT1,,OPTn).
• Two possible cases
• There is a bidder i such thatvi(M) OPT / m½.
• For all bidders vi(M) lt OPT / m½

Value
OPT/m½
2
3
4
1
Value
OPT/m½
We will provide a different O(m½)-mechanism for
each case. Later we will see how to combine them.
OPT2
OPT3
OPT4
OPT1
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The First Case (A Dominant Bidder) is Easy

• The second-price mechanism Bundle all items
  together. Assign the new bundle to bidder i that
  maximizes vi(M). Let the winner pay the second
  highest price.

Winner pays 40!
50
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40
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The Second Case (There is no Dominant Bidder)

• The fixed-price mechanism

• Define a per-item price pOPT / 2m
• For every bidder i1n
• Ask i for his most demanded bundle, Si, given the
  per-item price p.
• Allocate Si to i, and charge him pSi.

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The Second Case (No Dominant Bidder)
A
D
C
E
B
p
p
p
p
p
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The Second Case (No Dominant Bidder)
A
D
C
E
B
Blue bidder takes A,D and pays 2p.
p
p
p
p
p
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The Second Case (No Dominant Bidder)
C
E
B
Red bidder takes C and pays p.
p
p
p
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The Second Case (No Dominant Bidder)
E
B
Green bidder takes B,E and pays 2p.
p
p
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Proving the Approximation Ratio of the
Fixed-Price Auction (if there is no dominant
bidder)

• The fixed-price auction is clearly truthful.
• Lemma If for each bidder i, vi(OPTi) lt OPT/m½,
  then we get an O(m½)-approximation.
• Proof
• Claim Let PROFITABLEi vi(OPTi) p OPTi
  gt 0. Then, Si?PROFITABLE vi(OPTi) gt OPT/2.
• Informally, this means that most bundles in OPT
  are profitable given a fixed item-price of p.

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Proving the Approximation Ratio of the
Fixed-Price Auction (if there is no dominant
bidder)

• Proof (of claim)
• Si?N \ PROFITABLE vi(OPTi) lt Si?N \ PROFITABLE p
  OPTi (OPT / (2m) ) m OPT / 2

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Proving the Approximation Ratio of the
Fixed-Price Auction (if there is no dominant
bidder)

• If the mechanism gets to bidder i?PROFITABLE, and
  all items in OPTi are unassigned then bidder i
  will purchase at least one item.
• Whenever we sell a bundle S to bidder i, we gain
  a revenue of Sp. Clearly, vi(S) gt Sp S
  OPT/(2m).
• In the worst case, each item j?S is given to a
  different bidder in OPT. Hence, we lose
  (compared to OPT) at most SOPT / (m½) by
  assigning the items in S to i. We also lose a
  value of at most OPT / (m½) by not assigning i
  the bundle OPTi.
• This leads to a O(m½)-approximation to the social
  welfare of the bidders in PROFITABLE (gt OPT/2).

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Choosing between the Second-Price Auction and the
Fixed-Price Auction

• We flip a random coin.
• With probability ½ we run the second-price
  auction, and with probablity ½ we run the
  fixed-price auction.
• Still truthful.
• Still Guarantees the approximation ratio (in
  expectation).

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Getting Rid of the Assumption

• It is hard to estimate the value of OPT
• Recall that any approximation better than m½
  requires exponential communication.
• Estimating OPT requires information from the
  bidders.
• We use the optimal fractional solution instead.
• We get the information in a careful way.

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The Linear Relaxation

• Maximize Si,Sxi,Svi(S)
• Subject To
• For each item j Si,Sj?Sxi,S 1
• For each bidder i SSxi,S 1
• For each i,S xi,S 0
• Despite the exponential number of variables, the
  LP relaxation can still be solved in polynomial
  time using demand oracles (Nisan-Segal).
• OPTSi,Sxi,Svi(S) is an upper bound on the value
  of the optimal integral solution.

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Two Possible Cases

• Two possible cases
• ? bidder i such that vi(M) OPT / m½.
• For all bidders vi(M) lt OPT/m½.

Value
OPT/m½
OPT2
OPT3
OPT4
OPT1
The mechanism for the first case remains the
same.
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The Second Case (No Dominant Bidder)

• The key observation A randomly chosen set, that
  consists of a constant fraction of the bidders,
  holds (w.h.p.) a constant fraction of the total
  social welfare.
• This idea is similar to the main principle in
  random-sampling auctions for digital goods.
  Fiat-Goldberg-Hartline-Karlin-Wright
• By partitioning the bidders into two sets of
  equal size, we can use one set to gather
  statistics that will determine the per-item price
  of the other.

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The Second Case (No Dominant Bidder)

• The mechanism
• Randomly partition the bidders into two sets of
  size n/2 FIXED and STAT.
• Calculate the optimal fractional solution for
  STAT, OPTSTAT.
• Conduct a fixed-price auction on the bidders in
  FIXED with a per-item price of pOPTSTAT/(2m).

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Proving the Approximation Ratio of the
Fixed-Price Auction (if there is no dominant
bidder)

• The mechanism is clearly universally truthful.
• Theorem If for each bidder i, vi(M)lt(OPT/m1/2)
  then the fixed-price auction guarantees an
  O(m1/2)-approximation.
• Claim With probability 1-o(1) it holds
  thatOPTSTAT OPT/4 andOPTFIXED OPT/4

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Proving the Approximation Ratio of the
Fixed-Price Auction (if there is no dominant
bidder)

• Corollary With high probability p OPT / (8m)
• Reminder p OPTSTAT / (2m) and OPTSTAT gt
  OPT/4
• Claim Let PROFITABLE(i ,S) i?FIXED and vi(S)
  pOPT gt 0.Then S(i,S)?PROFITABLE
  xi,Svi(Si) gt OPT / 8.

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Proving the Approximation Ratio of the
Fixed-Price Auction (if there is no dominant
bidder)

• Claim For each item we sell at price OPT /
  (8m), we lose a value of at most OPT / O(m½)
  compared to the total social welfare of the
  (fractional) bundles in PROFITABLE. Since
  S(i,S)?PROFITABLE vi(S) gt OPT/8, we obtain an
  O(m½)-approximation mechanism for this case (no
  dominant bidder).

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Final Improvement Increasing the Probability of
Success

• The expected value of the solution provided by
  the mechanism is indeed O(m½).
• However, it only succeeds if it guesses the
  correct case. This occurs with a probability of
  ½.
• Success probability can be increased by running
  both mechanisms and choosing the allocation with
  the maximal value, or by using amplification.
  However, truthfulness is not preserved.
• Theorem For any egt0, there exists a truthful
  mechanism that achieves an O(m½ /
  e3)-approximation with probability 1-e.

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A Truthful Mechanism for General Valuations

• Phase I Partitioning the BiddersRandomly
  partition the bidders into three sets SEC-PRICE,
  FIXED, and STAT, such that SEC-PRICE(1-e)n,
  FIXED(e/2)n, and STAT(e/2)n.
• Phase II Gathering StatisticsCalculate the
  value of the optimal fractional solution in the
  combinatorial auction with all m items, but only
  with the bidders in STAT. Denote this value by
  OPTSTAT.
• Phase III A Second-Price Auction Conduct a
  second-price auction with a reserve price for
  selling the bundle of all items to one of the
  bidders in SEC-PRICE. Set the reserve price to be
  (OPTSTAT/m1/2). If there is a winning bidder
  allocate all the items to him. Otherwise, proceed
  to the next phase.

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A Truthful Mechanism for General Valuations

• Phase IV A Fixed-Price Auction Conduct a
  fixed-price auction with the bidders in FIXED and
  a per-item price of p(eOPTSTAT/8m).

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Correctness of the Final Mechanism

• If there is a dominant bidder i, then he will
  be in SEC_PRICE with probability 1-e.
• With probability of at most e the mechanism
  fails.
• Since OPTSTAT OPT the reserve price is at
  most OPT / m½.
• Therefore, we will have a winner in the
  second-price auction. The social welfare value we
  achieved is at least vi(M) gt OPT / m½.

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Handling the Case when there is no Dominant Bidder

• Claim With probability 1-o(1) it holds that
  OPTSTAT OPT/ 4e and OPTFIXED OPT / 4e
• With probability of at most o(1) the mechanism
  fails
• If there is a winner in the second-price auction
  then we are done.
• Otherwise, we have a good estimation of OPT (up
  to O(e)), and the fixed-price auction will
  provide a good approximation to the total social
  welfare.

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Other Results

• Using the same general framework we design a
  universally truthful O(log2m)-approximation
  mechanism for combinatorial auctions with XOS
  bidders.
• The XOS class includes all submodular valuations.
• Submodular v(S?T) v(S??T) v(S) v(T).
• Semantic Characterization Decreasing Marginal
  Utilities.

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Open Questions
Designing a truthful deterministic mechanism for
combinatorial auctions that obtains a O(m1/2)
approximation ratio.
computationally achievable
truthful approximations
Submodular valuations
e/(e-1)-e (Feige, Vondrak)

• log2m

Complement Free valuations

• m1/2(Dobzinski-Nisan-Schapira)

2 (Feige)