Approximating optimal combinatorial auctions
for complements
using restricted welfare maximization
Pingzhong Tang and Tuomas Sandholm
Computer Science Department
Carnegie Mellon University
High-level contributions
• New approach to mechanism design: “Social welfare with holes”
– I.e., curtail the set of allocations based on agents’ reports (e.g., bids), and
use welfare maximization within remaining set
– Unlike maximum-in-range approach [Nisan and Ronen 07], where the
allocation set is curtailed ex ante
– Completely general (e.g., remove all but one allocation)
– Trickier because not all report-based ways of curtailing are incentive
compatible (paper contains an example)
– We present the first (non-trivial) such curtailing that maintains incentive
compatibility
– Hopefully, a fruitful avenue going forward
• New, general form of reserve pricing for combinatorial auctions
– Any efficient mechanism can be arbitrarily far from optimal revenue,
while our reserves avoid this downside
Background
• Optimal (i.e., expected revenue maximizing) auctions known for:
– Single item [Myerson 81]
– Multiple identical units [Maskin and Riley 89]
– Multiple items with complementarities in a 1-dimensional setting [Levin 97]
• These are all based on virtual welfare maximization
– Requires prior information
– Complex and unintuitive
– Inefficient
• Welfare maximizing allocation rule, but with reserve prices
– Symmetric (1-item) setting: Identical to Myerson
– Asymmetric (1-parameter) setting: 2-approximation [Hartline and
Roughgarden EC-09]
Our technical contributions
• We approximate Levin's optimal auction for complements using welfare
maximization with a form of reserve pricing for combinatorial auctions
– Reserve prices restrict allocations based on bids
– In Levin's setting, we use a specific form: Monopoly reserves
– We obtain a 2-approximation to optimal revenue
• And a 6-approximation using anonymous reserves
• Why are we doing this?
– More efficient than Levin’s auction
• Any efficient mechanism has arbitrarily low revenue (e.g., in the paper)
– Requires less info to verify correct execution of the mechanism (given the
reserves)
– Simpler, easier to understand
– Better starting points for automated mechanism design (than, e.g., VCG)
Myerson's setting
• Seller has 1 indivisible item for sale, which he values at 0
• Set of bidders 1,…,n
– Bidder i’s valuation, vi , is private knowledge
– Distribution Fi and regular density fi, according to which vi is drawn, are
common knowledge
– Quasi-linearity: ui = vi – paymenti
• Two constraints:
– (Ex interim) incentive compatibility
– (Ex interim) individual rationality
• Objective: Maximize seller's expected revenue
Myerson's solution
• Asymmetric case: fi’s are different
– Virtual valuation:
– Allocation rule: Give the item to the bidder with the
highest virtual valuation, if it is positive and retain the item
otherwise
– Payment rule: The lowest bid by i that would have won
– Interpretation: Run second price auction on the virtual
valuations, with reserve price 0
• Symmetric case: fi = fj
– Optimal auction is a 2nd-price auction with monopoly
reserve price
– Monopoly reserve price: vri such that
More about asymmetric case…
• 2nd-price auction with monopoly reserve prices (one
per bidder) is a 2-approximation of Myerson's optimal
auction [Hartline and Roughgarden EC-09]
• A key step: Myerson's Lemma [1981]
– Lemma: For any truthful 1-item auction, expected payment
from a bidder equals his expected virtual valuation
Levin's setting:
Complements with 1-dimensional type
• Seller has 2 items for sale, which he values at 0
– All his results (and ours) extend to m items
• Set of bidders 1,…,n
– Bidder i’s type θi is private knowledge
– Distribution Fi and regular density fi, according to which θi is
drawn, are common knowledge
– Bidder i 's valuation function is
v
– Quasi-linearity: ui() = vi() - paymenti
• Two constraints:
– (Ex interim) incentive compatibility
– (Ex interim) individual rationality
• Objective: Maximize seller's expected revenue
Levin's solution
• Virtual valuation:
• Allocation Rule: Maximize virtual social welfare,
among all the positive virtual valuations
• Payment rule:
Case I: Agent i wins item 1 first:
Pay 0,
get nothing
Pay vi1(θ0),
get item 1
θ0
Pay additional vi2(θ1)+ vi3(θ1),
get both items
θ1
Case II: Agent i wins item 2 first:
Pay 0,
get nothing
Pay vi2(θ0),
get item 2
θ0
Pay additional vi1(θ1)+ vi3(θ1),
get both items
θ1
Approximating Levin's auction
• Why difficult?
– Multiple definitions of reserve prices in combinatorial settings
• One fake bidder, two fake bidders, bidder-specific...
• What are monopoly reserves in combinatorial settings?
– Myerson's Lemma in this setting?
– [Hartline and Roughgarden EC-09] approach doesn't apply
Our allocation-curtailing approach
applied to Levin's setting
• Idea
– Preclude bidder-bundle pairs that have negative virtual
valuations
– Preclude bidder-bundle pairs where removing some
item(s) from a bidder gives that bidder higher virtual value
• Theorem
– Together with welfare-maximization allocation rule and
Levin's payment rule, the preclusions above constitute an
auction that
• is incentive compatible (in weakly dominant strategies),
• is individually rational, and
• 2-approximates Levin's revenue
Desirable properties of our auction
• Incentive compatible, individually rational, 2-approximation
– Important step for proving this is allocation monotonicity:
Fixing others' reports, a bidder's set of allocated items is
expanding in his report
• More efficient than Levin
– Less restriction of the allocation space
– Welfare maximizing in this less restricted space
• Requires less information, e.g., to verify correct execution
– 5 numbers versus distribution function
• Easier to understand
• A bidder in his lowest type gets zero payoff
• For any allocation, a bidder's payment plus his virtual
valuation is no less than his real valuation
– We use this in 2-approximation proof
Extending Myerson's Lemma to this setting
• Myerson's Lemma: Bidder’s expected payment equals
his expected virtual value
• Our conditions:
– 1. Truthful
– 2. Allocation monotonic
– 3. Lowest type gets zero payoff
• Our auction satisfies 1, 2 and 3
• Levin's conditions:
– a. Truthful
– b. Revenue-maximizing
– c. Utility functions satisfy the requirements of envelope
theorem
Proof of 2-approximation
• Let M be the social welfare maximizing mechanism under monopoly
reserves (i.e, our auction)
• Step 1. By definition, M maximizes restricted social welfare
• Step 2. By Myerson’s lemma extended to this setting,
expected revenue of M = expected sum of bidders’ virtual valuations in M
• Step 3. As we prove, in M, a bidder's payment plus his virtual valuation is
no less than his real valuation
• Step 4. By Steps 2 and 3,
2 * [Expect revenue of M] ≥ social welfare of M
• Step 5. By Steps 1 and 4,
2 * [Expect revenue of M] ≥ social welfare of Levin
• Step 6. By individual rationality,
social welfare of Levin ≥ revenue of Levin
QED
6-approximation of Levin’s optimal revenue
using anonymous reserves
• Now, usual definition of reserve price:
– Seller pretends to have valuation a for 1st item, b for 2nd item, and c for
bundle
• Auction L: Levin's optimal auction on original set of bidders
• Auction D: Duplicate each bidder. Then apply welfare-maximizing
allocation rule and Levin payment rule
• Step 1. Auction D 3-approximates Auction L
• Step 2. Let a, b, and c be random variables that simulate maxi{vi1},
maxi{vi2} and maxi{vi1 + vi2 + vi3}, respectively, in the original bidder set
• Step 3. Step 2 trivially yields a 2-approximation of D. Hence, a 6approximation of L
QED
• In contrast to 4-approximation for 1-item setting [Hartline &
Roughgarden EC-09]
Conclusions
• New general approach to mechanism design: Social welfare with
holes
• New general form of reserve pricing under welfare maximization
in combinatorial auctions
• Application of this idea to Levin's setting of 1-D complements:
– 2-approximation to revenue
• 6-approximation with anonymous reserves
–
–
–
–
More efficient than Levin
Requires less info to verify correct execution (given reserves)
Easier to understand
Extended Myerson’s lemma to this setting
Future work
• Characterizing truthful restrictions
– 1-item setting: Equivalent to allocation monotonicity
– Levin's setting:
• In our follow-on work we have found a necessary condition (e.g.,
can go from nothing to winning Item 1 to winning Item 2 to
winning both)
• Plan to search for optimal auctions under this condition
• Application to other settings
• Application to automated mechanism design