Shahar Dobzinski and Shaddin Dughmi just posted a new paper to the arXiv: On the Power of Randomization in Algorithmic Mechanism Design.
The mechanism design problem that they study in this paper is the basic one of a multi-unit auction: There are identical indivisible units of some good for sale among bidders, and each bidder has a valuation function , where specifies his value from acquiring units. The normal assumptions are free disposal i.e. that the valuation functions are monotone non-decreasing. The goal is to find an allocation that gives units to each each bidder (and thus ) optimizing the social welfare .
This problem is perhaps the simplest one exhibiting the key clash between computational complexity and incentive compatibility: since this is a social welfare maximization problem, the VCG mechanism that finds the optimal allocation is incentive compatible. However, if we view the number of units as large, and require algorithms that work in time polynomial in and , (assuming that the valuation functions are either concisely represented or are given as “black boxes”) then finding the optimal allocation is a knapsack problem and is thus computationally hard. On the other hand, it is well known that the knapsack problem can be well-approximated efficiently (there is an FPTAS). The clash is that we don’t know how to turn such an approximation algorithm into an incentive-compatible approximation mechanism.
This paper manages to find a randomized incentive-compatible approximation mechanism. It also highlights the role and necessity of randomization in this setting.