Multiple Unit Auctions with Budget Constraint

Problem Definition

In this problem, an auctioneer would like to sell an idiosyncratic commodity with m copies to n bidders, denoted by ${ i=1,2, \dots ,n }$. Each bidder i has two kinds of privately known information: ${ t_i^u \in \mathbb{R^+} }$, ${ t_i^b \in \mathbb{R^+} }$. ${ t_i^u \in \mathbb{R^+} }$ represents the price buyer i is willing to pay per copy of the commodity and ${ t_i^b \in \mathbb{R^+} }$ represents i's budget.   

Then a one-round sealed-bid auction proceeds as follows. Simultaneously all the bidders submit their bids to the auctioneer. When receiving the reported unit value vector ${ \mathbf{u}=(u_1, \dots, u_n) }$ and the reported budget vector ${ \mathbf{b}=(b_1, \dots, b_n) }$ of bids, the auctioneer computes and outputs the allocation vector ${ \mathbf{x}=(x_1, \dots, x_n) }$ and the price vector ${ \mathbf{p}=(p_1, \dots, p_n) }$. Each element of the allocation vector indicates the number of copies allocated to the...