Truthfulness and Approximation

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

s

a

With g goods, the greedy allocation is within a factor of from the optimalg

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

gsk 2

kg2

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.