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.
During busy political seasons, many opinion polls are conducted. In apresidential race, how do you think the participants in polls are generally selected?Discuss any issues regarding simple random, stratified, systematic, cluster, andconvenience sampling in these polls. What about other types of polls, besides political?
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