52 Algorithmic Mechanism Design Problelns Problem 4.1 Consider a variant of a knapsack auction in which both the valuation v.1- and the size mg- of each bidder 3' are private. A mechanism now receives both bids b and reported sizes a from the bidders. An allocation rule 141(bg a) now species the amount of capacity allocated to each bidder} as a function of the bids and reported sizes. Feasibility dictates that 2?:1 331'\"): a) 3 W' for every b and a, where W is the total capacity of the shared resource. We dene the utility of a bidder 1' as eg- p.3-(b, a) if she gets her required capacity (i.e.} zi(b,a) 2 mi) and as pi(b,a) otherwise. This is not a singleparameter environment. Consider the following mechanism. Given bids b and reported sizes a} the mechanism runs the greedy knapsack auction of Sec tion 4.2.2, taking the reported sizes a at face value} to obtain a subset of winning bidders and prices 1). The mechanism concludes by award ing each Winning bidder capacity equal to her reported size (1.1-, at a price of pi; losing bidders receive and pay nothing. Is this mechanism DSlC? Prove it or give an explicit counterexample