Chapter 3, Problem 72E

Asymmetric method of sealed bids. Suppose that an estate consisting of M indivisible items is to be divided among N heirs $\left(P_1,P_2,\mathrm{...},P_N\right)$ using the method of sealed bids. Suppose that Grandma’s will stipulates that $P_1$ is entitled to $x_1\%$ of the estate, $P_2$ is entitled to $x_2\%$ of the estate,…, $P_N$ is entitled to $x_N\%$ of the estate. The percentages add up to 100%, but they are not all equal (Grandma loved some grandchildren more than others). Describe a variation of the method of sealed bids that ensures that each player receives a “fair share” (i.e., $P_1$ receives a share that she considers to be worth at least $x_1\%$ of the estate, $P_2$ receives a share that he considers to be worth at least $x_2\%$ of the estate, etc.).