Truthfulness and Approximation Kevin Lacker. Combinatorial Auctions Goals – Economically efficient...
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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, O’Callaghan, 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 (Gormak’s 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
s
a
With g goods, the greedy allocation is within a factor of from the optimalg
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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
[Mu’alem, 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 X’s 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
gsk 2
kg2
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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, O’Callaghan, Shoham. Truth Revelation in Approximately Efficient Combinatorial Auctions.
Mu’alem, Nisan. Truthful Approximation Mechanisms for Restricted Combinatorial Auctions.