Simplified Abstraction

In the simple model, there are $n$ items to be auctioned off in sequence (first item 0, then item 1, etc.). The bidder must place a bid $r_i$ for each item $i$, and after each bid, a closing price $y_i$ is chosen for the corresponding item from a distribution specific to the item. If the bid matches or exceeds the closing price, $r_i \ge y_i$, the bidder holds item $i$, $h_i=1$. Otherwise, the bidder does not hold the item, $h_i=0$. The bidder's utility $v(H)$ is a function of its final vector of holdings $H=(h_0,\ldots,h_{n-1})$ and its cost is a function of the holdings and the vector of closing prices, $H \cdot Y$. We will formalize the problem of optimal bid selection and develop a series of approximations to make the problem solvable.