The paper Incentive Compatible Budget Elicitation in Multi-unit Auctions by Sayan Bhattacharya, Vincent Conitzer, Kamesh Munagala, and Lirong Xia has recently been uploaded to the arXiv.
The basic model that they study is the auction of a single infinitely divisible good among bidders with budget limits. In the model each bidder has a private value and private budget . The utility of when allocated units of the infinitely divisible good and charged is assumed to be but only if the budget is not exceeded, . If then the utility is assumed to be negative infinity.
The twist in this model is the budget limit which takes us out of the quasi-linear model usually assumed in auction theory. In particular VCG mechanisms are no longer incentive compatible. A previous paper by Shahar Dobziniski, Ron Lavi, and myself shows that indeed there is no (anonymous) incentive compatible mechanism that produces Pareto-optimal allocations. This paper shows that there is such a randomized mechanism.
The paper starts with the mechanism that was previously shown to be incentive compatible in the special case that the budgets are public knowledge (an adaptive version of Ausubel’s auction). The heart of the proof lies in an analysis showing that the only way a bidder can gain by mis-reporting his budget is by over-reporting; under-reporting is never profitable. This requires quite a delicate analysis (which I didn’t fully read yet) but once this characterization is known, an incenive compatible randomized mechanism follows immediately: run the original mechanism, and instead of asking a bidder to pay , ask him to pay with probability . The expected payments have not changed so under-reporting is still not profitable, and this randomized payment makes over-reporting result in a postive probability of getting negative infinite utility and thus is certainly not profitable.