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 ( s 1 , s 2 , .... , s 6 ) . The partnership is splitting up, and the partners are going to divide fairly the six parcels among themselves. T a b l e 3 − 2 9 shows the value of each parcel in the eyes of each partner. T a b l e 3 − 2 9 <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.
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 ( s 1 , s 2 , .... , s 6 ) . The partnership is splitting up, and the partners are going to divide fairly the six parcels among themselves. T a b l e 3 − 2 9 shows the value of each parcel in the eyes of each partner. T a b l e 3 − 2 9 <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.
Solution Summary: The author explains that an efficient fair division of the six parcels among the three partners is called envy-free.
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
(
s
1
,
s
2
,
....
,
s
6
)
.
The partnership is splitting up, and the partners are going to divide fairly the six parcels among themselves.
T
a
b
l
e
3
−
2
9
shows the value of each parcel in the eyes of each partner.
T
a
b
l
e
3
−
2
9
<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 notenvy-free.
d. Find a fair division of the six parcels among the three players that is envy-free but not efficient.
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