Game Development Reference
In-Depth Information
The decision of the second person is the more obvious as long as we assume
that the person is going to attempt to get as much cake as he can. (I don't think
that's a far stretch, do you?) The key point—and therefore the more interesting de-
cision—falls to the first person—the Cutter. Where does the Cutter divide the cake
so that he can also maximize the amount of cake he would get?
I am going to put off the obvious solution for a moment so that we can formally
approach the logic involved. The Cutter has the power to do one of three things
(ignore the symmetry for the moment). He can make piece A bigger (and B smaller,
of course), he can make piece B bigger (at the expense of A), or he can do his darn-
dest to make them the same size. If he makes A bigger, the Decider is likely to select
A. If he makes B bigger, the Decider is, once again, likely to select B. In both cases,
the Cutter comes out on the short end of the spatula and gets a smaller piece.
However, as the Cutter minimizes the difference between the two pieces—even to
the point of theoretical equality—the Decider loses the ability to even
detect
which
of the two pieces is bigger, much less to select it.
Even if the Decider could tell the difference down to miniscule amounts, as the
size of the larger piece approaches the size of the Cutter's inevitably smaller piece,
that smaller piece is increasing in size as well. If we assume that the meticulous mea-
surements of the Decider will
always
allow him to pick the larger piece, the best the
Cutter can hope for is half the cake minus some nanoscale amount. Obviously, it is
in the Cutter's best interest to make that amount as small as possible since it is the
only thing standing between him and a complete half of the cake.
From a game theory standpoint, the only way the Cutter can determine his
strictly dominant strategy is to take into account what the Decider may do given
each of the possible ways of cutting the cake (Figure 5.6). So, like the Prisoner's
Dilemma, the strictly dominant strategy is to attempt to maximize your own posi-
tion. Unlike the Prisoner's Dilemma, however, the optimum strategy is the same as
the dominant one. You simply can't improve the dominant strategy any further.
The main similarity to the Prisoner's Dilemma, however, is that the key to
finding that strategy is to think beyond your own position and put yourself into the
mind of the opponent. That is, “How would the other person react to the choice I
make?� If you didn't have to take this into consideration, the “game� would be as
simple as taking as much cake as you like and leaving the rest for the other person.
After all, they wouldn't have a choice but to accept what you offered them.
In late 2007, a rumor began circulating claiming that “the cake is a lie.� I can
assure you that this is, in no way, a reference to the cake in this example. Any
attempt by nefarious sorts to imply that the cake in this example is a falsehood
or other such fabrication will be dealt with swiftly.
