ECONOMY

426 fair division
Some of the cake-cutting procedures that have been proposed are discrete, whereby
players make cuts with a knife—usually in a sequence of steps—but the knife is not
allowed to move continuously over the cake. Moving-knife procedures, on the other
hand, permit such continuous movement and allow players to call “stop” at any point
at which they want to make a cut or mark.
There are now about a dozen procedures for dividing a cake among three players,
and two procedures for dividing a cake among four players, such that each player is
assured of getting a most valued or tied-for-most-valued piece (Brams, Taylor, and
Zwicker 1995; Barbanel and Brams 2004). When a cake can be so divided, no player
will envy another player, resulting in an envy-free division.
In the literature on cake-cutting, two assumptions are commonly made:
1. The goal of each player is to maximize the minimum-valued piece (maximin
piece) he or she can guarantee for himself or herself, regardless of what the other
players do. To be sure, a player might do better by not following such a maximin
strategy; this will depend on the strategy choices of the other players. However,
all players are assumed to be risk averse: They never choose strategies that might
yield them larger pieces if they entail the possibility of giving them less than their
maximin pieces.
2. The preferences of the players over the cake are continuous. Consider a procedure
in which a knife moves across a cake from left to right and, at any moment,
the piece of the cake to the left of the knife is A and the piece to the right is B.
The continuity assumption enables one to use the intermediate value theorem
to say the following: if, for some position of the knife, a player views piece A as
being larger than piece B, and for some other position he or she views piece B as
being larger than piece A, then there must be some intermediate position such
that the player values the two pieces exactly the same.
Only two three-person procedures (Stromquist 1980; Barbanel and Brams 2004),
and no four-person procedure, make an envy-free division with the minimal number
of cuts (n − 1cutsiftherearen players). A cake so cut ensures that each player gets a
single connected piece, which is especially desirable in certain applications (e.g. land
division).
For two players, the well-known procedure of “I cut the cake, you choose a piece,”
or “cut-and-choose,” leads to an envy-free division if the players choose maximin
strategies. The cutter divides the cake 50–50 intermsofhisorherpreferences.
(Physically, the two pieces may be of different size, but the cutter values them the
same.) The chooser takes the piece he or she values more and leaves the other piece
for the cutter (or chooses randomly if the two pieces are tied in his or her view).
Clearly, these strategies ensure that each player gets at least half the cake, as he or she
values it, proving that the division is envy free.
But this procedure does not satisfy other desirable properties. For example, if the
cake is, say, half vanilla, which the cutter values at 75 per cent, and half chocolate,
whichthechooservaluesat75 per cent, a “pure” vanilla–chocolate division would be
better for both players than the divide-and-choose division, which gives each player