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Truthfulness and Approximation. Kevin Lacker. Combinatorial Auctions. Goals Economically efficient Computationally efficient Problems Vickrey auction is hard Finding social optimum is hard Even just communicating your type is hard. Single Minded Bidders.
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Truthfulness and Approximation Kevin Lacker
Combinatorial Auctions • Goals • Economically efficient • Computationally efficient • Problems • Vickrey auction is hard • Finding social optimum is hard • Even just communicating your type is hard
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]
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
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
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
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)
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
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}.)
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
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
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}.)
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
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
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]
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
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
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
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.
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
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
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}.)
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
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
Papers cited • Lehmann, O’Callaghan, Shoham. Truth Revelation in Approximately Efficient Combinatorial Auctions. • Mu’alem, Nisan. Truthful Approximation Mechanisms for Restricted Combinatorial Auctions.