Chapter 3, Problem 68E

Efficient and envy-free fair divisions. A fair division is called efficient if there is no other fair division that gives every player a share that is as good or better (i.e., any other fair division that gives some players a better share must give some other players a worse share). A fair division is called envy-free if every player ends up with a share that he or she feels is as good as or better than that of any other player.

Suppose that three partners (A, B, and C) jointly own a piece of land that has been subdivided into six parcels $\left(s_1,s_2,\mathrm{....},s_6\right).$ The partnership is splitting up, and the partners are going to divide fairly the six parcels among themselves. $Table3-29$ shows the value of each parcel in the eyes of each partner.

$Table3-29$

<AC>A</AC>		B	C
$s_1$	$20,000	$16,000	$19,000
$s_2$	$19,000	$18,000	$18000
$s_3$	$18,000	$19,000	$15,000
$s_4$	$16,000	$20,000	$12,000
$s_5$	$15,0000	$15,000	$20,000
$s_6$	$20,000	$12,000	$16,000

a. Find a fair division of the six parcels among the three players that is efficient.

b. Find a fair division of the six parcels among the three players that is envy-free.

c. Find a fair division of the six parcels among the three players that is efficient but not envy-free.

d. Find a fair division of the six parcels among the three players that is envy-free but not efficient.