Truthfulness and Approximation

1. Truthfulness and Approximation Kevin Lacker

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

3. 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]

4. 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

5. 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

6. 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

7. 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)

8. 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

9. 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}.)

10. 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

11. 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

12. 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}.)

13. 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

14. 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

15. 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]

16. 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

17. 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

18. 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

19. 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.

20. 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

21. 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

22. 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}.)

23. 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

24. 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

25. Papers cited • Lehmann, O’Callaghan, Shoham. Truth Revelation in Approximately Efficient Combinatorial Auctions. • Mu’alem, Nisan. Truthful Approximation Mechanisms for Restricted Combinatorial Auctions.