Sort and re-index the bidders so that bitvi > b2w2 > > brawn. Pick winners in this order until one doesn’t fit, and then halt. Return either the solution from the previous step or the highest bidder, whichever has larger social welfare. Call this allocation rule “Method X”. Does a payment rule exist to make Method X DSIC?

Imagine you are the captain of a cargo ship and different companies want you to transport their cargo. Each piece of cargo has a certain weight and valuation below. Suppose the ship can carry 17 tons of cargo. Item’s valuations are kept private by the companies but item weights are known publically.

Item/Bidder Valuation Weight

A                 10                10

B                  9                   8

C                  6                    5

D                 4.5                 5

(a) (3 pts) Run the Greedy Algorithm and compare that to the Social Welfare Maximizing Allocation.

(b) (5 pts) Determine the price bidder C will pay under the Greedy Algorithm.

(c) (5 pts) Sort and re-index the bidders so that bitvi > b2w2 > > brawn. Pick winners in this order until one doesn’t fit, and then halt. Return either the solution from the previous step or the highest bidder, whichever has larger social welfare. Call this allocation rule “Method X”. Does a payment rule exist to make Method X DSIC?

(d) (5 pts) For the Greedy Knapsack Heuristic, we are guaranteed to obtain at least 1/2 of the maximum social welfare. Discover the analogous guaranteed social welfare of the allocation rule of Method X or prove that no such guarantee exists. You may assume truthful bidding.