Two person fair division
The standard method for dividing a piece of cake: you cut I choose, is fair in the sense that it
is proportionate (i.e., each person gets at least 1/2 of the cake by their own evaluation) and
it is envy free (i.e., neither person would prefer the other person's portion). Yet it is not
fair, I would much prefer to have you cut, which gives me the choice, rather than cut the cake
and give you the choice. The reason is that cake is not homogeneous, your and my relative
preferences for cake, frosting, and aesthetic appearance may be different. The cutter divides
the cake into two portions which he perceives as equal, but the chooser may not perceive them as
equal. Hence the cutter gets half the cake (by his perception), but the chooser may get more
than half the cake (by his perception). For example, if a cake were half chocolate and half
white, and the cutter was indifferent to chocolate versus white cake, the cutter might cut the
cake down the middle producing one chocolate and one white piece; in that case, if the chooser
preferred chocolate cake, he would be much happier with the chocolate piece than with the white
piece.
Conversely, if the preferences of the chooser (or both
parties) are known, the advantage is shifted to the cutter. For example, if the chooser
prefers frosting, while the cutter likes both frosting and cake equally well, the cutter will
put enough excess frosting in the smaller portion to entice the chooser to take it. Hence the
chooser will get just over half the value of the cake (by the chooser's values) (the cutter must
make sure the chooser chooses the proper portion), while the cutter will get substantially over
half the value of the cake (by the cutter's values). Using the example of the cake which is
half chocolate and half white, if the cutter is indifferent between chocolate and white cake
but the chooser is allergic to chocolate (and the cutter knows that fact); the cutter could cut
the cake into 1/3 white (i.e., 2/3 of the white half) versus 1/6 white with 1/2 chocolate. The
chosser would choose the 1/3 white, and the cutter would end up with 2/3 of the original cake.
What do we mean by fair? If your mother bakes a cake which is half chocolate and half white,
and cuts it so that you and your sibling each get one-quarter chocolate cake and one-quarter
white cake, that is certainly fair. But if you prefer chocolate cake and your sibling prefers
white cake, you would both be happier if you had one-half chocolate cake and your sibling had
one-half white cake. The term Pareto optimal refers to divisions where no exchange could
make both parties happier. As long as you have some white cake and your sibling has some
chocolate cake, you would both benefit by an exchange, and the division is not Pareto optimal.
Although giving you all the chocolate and your sibling all the white is pareto optimal, so is
giving you the entire cake (giving up some of the cake will not make you happier), or giving
the entire cake to your sibling. Any division where either you have no white cake or your
sibling has no chocolate cake is Pareto optimal (you would require more more white cake than the
chocolate you gave up in an exchange, but such an exchange would not be acceptable to your
sibling).
Assume you like chocolate cake twice as much as white (i.e., you have equal preference for one
square of chocolate or two squares of white). Assume also that your sibling likes white cake
thrice as much as chocolate (i.e., your sibling has equal preference for one square of white or
three squares of chocolate). The Pareto optimal divisions are as described above (either you
have no white or your sibling has no chocolate). But what apportionment would ensue from
trading after your mother's "fair" division?
Since you like chocolate equal to twice as much white cake, you would be willing to give up your
1/4 white cake for one-eighth chocolate cake, resulting in your sibling having all the white
cake (one-half cake) and one-eighth chocolate cake. But your sibling would be willing to give
up chocolate cake for one-third as much white cake, so if you offered one-twelfth white cake
(1/3 of the 1/4 your mother gave you), you could get all your sibling's chocolate cake
(one-quarter cake) with the ultimate result you would have all the chocolate cake (one-half
cake) plus 1/6 white cake, while your sibling had only 1/3 white cake. Hence even with an
initial division, there will in general not be a unique Pareto optimal division which ensues
from it. How much you can get will depend on whether you know your siblings preferences, and
how good you are at bargaining. Any division between the two extremes calculated is possible.
Exercise: How would Jack Sprat divide a roast if he did not know his wife's
preferences? How would he divide the roast if he knew his wife's preferences?
If you knew
the chooser wanted exactly equal amounts of cake and frosting, how would you divide a cake?
If you preferred chocolate cake thrice as much as white, and your sibling preferred white cake
quatrice as much as chocolate, what are the Pareto optimal divisions of a cake which is 1/2
chocolate and 1/2 white? Which of those partitions could ensue from an initial division
giving each of you 1/4 chocolate and 1/4 white cake?
If the matter to be divided is
discrete rather than a continuum, the chooser has the obvious advantage. For example, if there
are 5 marbles to be divided, the fairest division is 2 and 3, and the chooser gets three. If the
objects are not the same, the advantage may not be so clear. For example, if there is a red
shirt, a green shirt, red pants, and green pants, the "cutter" decides according to his pleasure
whether the wardrobes will be mixed or matched; if the "cutter" decides on mixed wardrobes, the
chooser canot get green pants with a green shirt.
A more common implementation is
alternating choices, in which case the first choice is a definite advantage (unless compensated
by the next two choices going to the second individual, or some other caompensatory
mechanism).
If you know your
opponents preferences, you may have a significant advantage under alternate choices. Assume a
family estate consists of A tailcoat (T), a wedding gown (W), a pair of cufflinks ((C), a
necklace (N), a samovar (S), and a mirror (M).Annabelle has the first choice, and the preference
order WNSMTC, Alfred has the second choice and the preference TCSMNW. Aternating choices should
provide Annabelle with WNS and Alfred with TCM. But if Alfred knows Annabelle does not want
cufflinks, he can put them at the bottom of his preference and end up with TSC while Annabelle
gets WNM. (Of course if Annabelle knew Alfred's preferences, she could put the wedding gown at
the bottom of her list ... .)
Competencies:
Reflection: Under what
circumstances is cutting versus choosing an advantage?.
Challenge:
February 2004
return to index
campbell@math.uni.edu