Incentive-compatible Approximation Andrew Gilpin 10/25/07

IC Approximation Most mechanism design solutions are not computationally efficient (e.g. VCG) –Therefore not implementable Most algorithms (exact or approximate) are not incentive-compatible (IC) –Therefore do not achieve economic goals How can we design IC approximation algorithms?

Overview of today Job scheduling –Single-dimensional domain –Different goals of algorithms –Designing IC algorithms Combinatorial auctions –Multi-dimensional domain –Multi-dimensional monotonicity handled by randomness –Alternative concepts of “truthfulness”

Motivating example: “Approximate VCG” not IC Items: {A,B,C} Bidder 1:,, Bidder 2:,, ????

Job scheduling n jobs are to be assigned to m machines Job j consumes p j time-units Machine i has speed s i –So machine i requires p j / s i time-units to complete job j Let l i = Σ j | j is assigned to i p j be the load on machine I Goal: minimize max i l i / s i

Utility in the job scheduling domain The machines are the agents Each machine incurs a unit cost for every consumed time unit Utility: u(l i,P i ) = -l i / s i - P i

Can we just apply VCG? Recall objective: minimize max i l i / s i The objective in VCG is to minimize the sum of costs (welfare) Furthermore, computing the optimal outcome is NP-hard

General definition

Monotone algorithm for job shop scheduling: Randomized algorithm

So far… We have a polynomial time algorithm that gives a 2-approximation It is only truthful in expectation What about truthfulness?

A truthful deterministic algorithm Open question: Does there exist a truthful PTAS for scheduling related machines?

Multidimensional domains: Combinatorial auctions

Transition to integral case

Decomposition technique Primal has exponentially many variables Dual has polynomially many variables, but exponentially many constraints Dual has a poly-time seperation routine –Thus, Dual can be solved by Ellipsoid Algorithm

Verifying integrality gap

Open questions Combinatorial auctions with submodular valuations are NP-hard but have allow a constant-factor approximation. Does there exist a truthful constant-factor approximation? What (precisely) are the limitations of deterministic truthful combinatorial auctions?

Final comment about combinatorial auctions All of this has focused on welfare maximization. Very little is known about revenue maximization.