91Ó°ÊÓ

Henry, Tom, and Fred are dividing among themselves a set of common assets equally owned by the three of them. The assets are divided into three shares \(\left(s_{1}, s_{2},\right.\) and \(\left.s_{3}\right) .\) Table 12 shows the values of the shares to each player expressed as a percent of the total value of the assets. (a) Which of the shares are fair shares to Henry? (b) Which of the shares are fair shares to Tom? (c) Which of the shares are fair shares to Fred? (d) Find all possible fair divisions of the assets using \(s_{1}, s_{2}\), and \(s_{3}\) as shares. (e) Of the fair divisions found in (d), which one is the best? $$ \begin{array}{|l|c|c|c} \hline & s_{1} & s_{2} & s_{3} \\ \hline \text { Henry } & 25 \% & 40 \% & 35 \% \\ \hline \text { Tom } & 28 \% & 35 \% & 37 \% \\ \hline \text { Fred } & 33 \frac{1}{3} \% & 33 \frac{1}{3} \% & 33 \frac{1}{3} \% \end{array} $$

Allen, Brady, Cody, and Diane are sharing a cake. The cake had previously been divided into four slices \(\left(s_{1}, s_{2}, s_{3},\right.\) and \(\left.s_{4}\right)\). Table 16 shows the values of the slices in the eyes of each player. (a) Which of the slices are fair shares to Allen? (b) Which of the slices are fair shares to Brady? (c) Which of the slices are fair shares to Cody? (d) Which of the slices are fair shares to Diane? (e) Find all possible fair divisions of the cake using \(s_{1}, s_{2}, s_{3}\), and \(s_{4}\) as shares. $$ \begin{array}{|c|c|c|c|c|} \hline & s_{1} & s_{2} & s_{3} & s_{4} \\ \hline \text { Allen } & \$ 4.00 & \$ 5.00 & \$ 6.00 & \$ 5.00 \\ \hline \text { Brady } & \$ 3.00 & \$ 3.50 & \$ 4.00 & \$ 5.50 \\ \hline \text { Cody } & \$ 6.00 & \$ 4.50 & \$ 3.50 & \$ 4.00 \\ \hline \text { Diane } & \$ 7.00 & \$ 4.00 & \$ 4.00 & \$ 5.00 \\ \hline \end{array} $$

Anne, Bette, and Chia jointly own a flower shop. They can't get along anymore and decide to break up the partnership using the method of sealed bids, with the understanding that one of them will get the flower shop and the other two will get cash. Anne bids \(\$ 210,000,\) Bette bids \(\$ 240,000,\) and Chia bids \(\$ 225,000 .\) How much money do Anne and Chia each get from Bette for their third share of the flower shop?

Consider the following method for dividing a continuous asset \(S\) among three players (two dividers and one chooser): Step 1. Divider \(1\left(D_{1}\right)\) cuts \(S\) into two pieces \(s_{1}\) and \(s_{2}\) that he considers to be worth, \(\frac{1}{3}\) and \(\frac{2}{3}\) of the value of \(S\), respectively. Step 2. Divider \(2\left(D_{2}\right)\) cuts the second piece \(s_{2}\) into two halves \(s_{21}\) and \(s_{22}\) that she considers to be of equal value. Step3. The chooser \(C\) chooses one of the three pieces \(\left(s_{1}\right.\), \(s_{21},\) or \(\left.s_{22}\right), D_{1}\) chooses next, and \(D_{2}\) gets the last piece. (a) Explain why under this method \(C\) is guaranteed a fair share. (b) Explain why under this method \(D_{1}\) is guaranteed a fair share (c) Illustrate with an example why under this method \(D_{2}\) is not guaranteed a fair share.