Truthfulness and Approximation

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Truthfulness and Approximation

• Kevin Lacker

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Combinatorial Auctions

• Goals
• Economically efficient
• Computationally efficient
• Problems
• Vickrey auction is hard
• Finding social optimum is hard
• Even just communicating your type is hard

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Single Minded Bidders

• Restrict possible bidder types to make the
  problem easier
• Each bidder is only interested in one exact
  subset of the available goods
• Different from single-parameter

Lehmann, OCallaghan, Shoham 99
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The Problem Is Still Not Trivial

• Communicating your type is easy
• But Vickrey auctions still infeasible
• Maximal independent set reduces to social optimum
• Real world examples
• Pollutant permits

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Greedy Allocation

• Sort each bid using some prioritizing scheme
• Greedily accept bids that do not conflict with a
  higher priority bid
• Hopefully, priority correlates to the economic
  efficiency of the bid

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How to Prioritize

• A good idea bid-monotonicity
• Shrinking your set of desired goods should
  increase priority
• Increasing the money you would pay should
  increase priority
• Some bid-monotonic priority functions
• The average price per good you are offering
• Can penalize or reward bids with large sets

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Example

• Use average price per good to prioritize
• Anna values a at 20
• Ben values b at 5
• Gormak values a,b at 30
• Priority order is Anna, Gormak, Ben
• We give a to Anna and give b to Ben
• Social welfare is 25 (not optimal)

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Payment Schemes

• Clarke scheme
• Each bidder pays their bid, minus the amount they
  improved the social welfare
• Works for generalized Vickrey auctions
• Does not yield a truthful mechanism when we are
  not finding the social optimum

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Example, Continued

• We sold a to Anna for 20 and b to Ben for 5.
• Suppose Anna had not existed
• We would sell a,b to Gormak and social welfare
  increases to 30
• The Clarke scheme would thus charge Anna 25 for
  something she values at 20

(Anna 20 for a. Ben 5 for b. Gormak 30 for
a,b.)
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Conditions for Truthfulness

• Exactness
• Bidders get either the set they bid for, or
  nothing.
• Monotonicity
• Winning bids still win with more money or less
  items
• Critical
• Bidders only pay the lowest bid that would have
  won
• Participation
• The utility of a losing bidder is zero

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A Truthful Mechanism

• Use greedy allocation with a bid-monotonic
  priority function
• Guarantees exactness and monotonicity
• Winning bidders pay the lowest bid that still
  would have won
• Guarantees critical and participation
• Easy to calculate

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Example Payments

• Anna won due to a higher priority than Gormak
• Minimum winning priority 15 (Gormaks priority)
• So Anna pays 15
• Ben won by default, he pays nothing
• In a Vickrey auction, Gormak wins and pays 25

(Anna 20 for a. Ben 5 for b. Gormak 30 for
a,b.)
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Greedy Can Increase Profit

• Dan values d at 9
• Eve values e at 1
• Lupin values d,e at 20
• With greedy, Lupin wins and pays 18
• With Vickrey, Lupin wins and pays 10

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Theorem

• Let a bid for set s and amount a get priority

• With g goods, the greedy allocation is within a
  factor of from the optimal

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Known Single Minded Bidders

• A further restricted model
• The mechanism designer already knows what set of
  goods each agent is interested in
• Conditions of exactness, monotonicity, critical,
  and participation still imply truthfulness

Mualem, Nisan 02
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Bitonic Mechanisms

• A subset of mechanisms obeying the previous four
  conditions
• Such a mechanism is bitonic iff
• For losing bids, social welfare is non-increasing
• For winning bids, social welfare is
  non-decreasing
• Greedy is bitonic

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Example of Not Bitonic

• A mechanism with the condition If Player X bids
  0, then Players X and Y are excluded.
• Still obeys exactness, monotonicity, critical,
  participation.
• Social welfare increases when Xs bid increases,
  even though it may be a losing bid
• Note this mechanism makes no sense

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More Bitonic Mechanisms

• Exhaustive-k
• Search all possible combinations of k bids
• Pick the valid combination maximizing social
  welfare
• Linear Programming
• Relax the integrality constraint (a bid is either
  accepted, or not)
• Accept all bids that the LP decides to 100
  accept

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Combining Mechanisms

• Given mechanisms A and B, run both of them and
  pick the result maximizing social welfare.
• If A and B are bitonic, Max(A,B) is also bitonic.
• If A or B is not bitonic, Max(A,B) is not
  guaranteed to be a truthful mechanism.

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Max Needs Bitonic

• Example one object, bidders A, B, and C
• Mechanism M1 If C bids in
• 0,10) A wins
• 10,20) B wins
• 20,) C wins
• Mechanism M2 C wins
• In Max(M1,M2), C may be incentivized to lie so
  that M2 defeats M1

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Max Needs Known Mindedness

• Many objects but only two are cared about
• Anna wants a for 19
• Ben wants b for 5
• Gormak wants a,b for 22
• Mechanism M1 Greedy, rank by average price
• Mechanism M2 Greedy rank by average price but
  object a counts as 10 objects

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Max Needs Known Mindedness

• M1 priority Anna, Gormak, Ben
• Anna and Ben win, Anna pays 11, Ben pays 0
• M2 priority Ben, Gormak, Anna
• Anna and Ben win, Anna pays 0, Ben pays 2
• Ben has incentive to add goods to his basket
• Lower his priority so M2 allocates to Gormak
• Ben pays the lower cost of M1

(Anna 19 for a. Ben 5 for b. Gormak 22 for
a,b.)
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Approximation Theorems

• With g goods, fix k, let M be greedy. For a bid
  of amount a and set s, give it priority a only if
• Max(M, Exhaustive-k) approximates to within

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Approximation Theorems

• Multi-unit auction
• Many identical goods
• V is greedy, where priority is the bid amount.
• D is greedy, where priority is the average price
  per good in the bid.
• Max(V,D) is a 2-approximation

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Papers cited

• Lehmann, OCallaghan, Shoham. Truth Revelation in
  Approximately Efficient Combinatorial Auctions.
• Mualem, Nisan. Truthful Approximation Mechanisms
  for Restricted Combinatorial Auctions.