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.