do not possess a complete theory.
Neither do war, nor business competi-
tion, nor any of the other forms of
competitive activity in which we are
really interested.
In a game like ticktacktoe, with a
small number of moves, where each
player is in a position to contemplate
all possibilities and to establish a de-
fense against the best possible moves of
the other player, a complete theory of
the von Neumann type is valid. In such
a case, the
game must inevitably end
in a win for the first player, a win for
the second player, or a draw.
I
question strongly whether this
concept of the perfect game is a com-
pletely realistic one in the cases of
actual, nontrivial games. Great generals
like Napoleon and great admirals like
Nelson have proceeded in a different
manner. They have been aware not
only of the limitations of their op-
ponents in such matters as materiel
and personnel but equally of their
limitations in experience and in mili-
tary know-how. It was by a realistic
appraisal of the relative inexperience
in naval operations of the continental
powers as compared with the highly
developed tactical and strategic
corn-
petence of the British fleet that Nelson
was able to display the boldness which
pushed the continental forces off the
seas. This he could not have done had
he engaged in the long, relatively in-
decisive, and possibly losing conflict to
which his assumption of the best pos-
sible strategy on the part of his enemy
would have doomed him.
In assessing not merely the materiel
and personnel of his enemies but also
the degree of judgment and the amount
of skill in tactics and strategy to be
expected of them, Nelson acted on the
basis of their record in previous
conl-
bats. Similarly, an important factor in
Napoleon's conduct of his combat with
the Austrians in Italy was his knowl-
edge of the rigidity and mental limita-
tions of Wiirmser.
This element of experience should
receive adequate recognition in any
realistic theory of games. It is quite
legitimate for a chess player to play,
not against an ideal, nonexisting, per-
fect antagonist, but rather against one
whose habits he has been able to de-
termine from the record. Thus, in the
theory of games, at least two different
intellectual efforts must be made. One
is the short-term effort of playing with
a determined policy for the individual
game. The other is the examination of
a record of many games. This record
has been set by the player himself, by
his opponent, or even by players with
whom he has not personally played.
In terms of this record, he determines
the relative advantages of different
policies as proved over the past.
There is even a third stage of judg-
ment required in a chess game. This is
expressed at least in part by the length
of the significant past. The develop-
ment of theory in chess decreases the
importance of games played at a dif-
ferent stage of the art. On the other
hand, an astute chess theoretician may
estimate in advance that a certain
policy currently in fashion has become
of little value, and that it may be best
to return to earlier modes of play to
anticipate the change in policy of the
people whom he is likely to find as
his opponents.
Thus, in determining policy in
chess there are several different levels
of consideration which correspond in
a certain way to the different logical
types of Bertrand Russell. There is the
level of tactics, the level of strategy,
the level of the general considerations
which should have been weighed in
determining this strategy, the level in
which the length of the relevant
past-
the past within which these considera-
tions may be valid-is taken into ac-
count, and so on. Each new level
demands a study of a much larger
past than the previous one.
I have compared these levels with
the logical types of Russell concerning
classes, classes of classes, classes of
classes of classes, and so on. It may be
noted that Russell does not consider
statements involving all types as
significant. He brings out the futility
of such questions as that concerning
the barber who shaves all persons, and
only those persons, who do not shave
themselves. Does he shave himself? On
one type he does, on the next type he
does not, and so on, indefinitely. All
such questions involving an infinity of
types may lead to unsolvable paradoxes.
Similarly, the search for the best policy
under all levels of sophistication is a
futile one and must lead to nothing
but confusion.
These considerations arise in the
determination of policy by machines
as well as in the determination of
policy by persons. These are the ques-
tions which arise in the programming
of programming. The lowest type of
game-playing machine plays in terms
of a certain rigid evaluation of plays.
Quantities such as the value of pieces
gained or lost, the command of the
pieces, their mobility, and so on, can
be given numerical weights on a cer-
tain empirical basis, and a weighting
may be given on this basis to each
next play conforming to the rules of
the game. The play with the greatest
weight may be chosen. Under these
circumstances, the play of the machine
will seem to its antagonist-who can-
not help but evaluate the chess per-
sonality of the machine-a rigid one.
Learning Machines
The next step is for the machine
to take into consideration not merely
the moves as they occurred in the in-
dividual game but the record of games
previously played. On this basis, the
machine may stop from time to
time,
not to play but to consider what (linear
or nonlinear) weighting of the factors
which it has been given to consider
would correspond best to won games as
opposed to lost (or drawn) games.
On this basis, it continues to play with
a new weighting. Such a machine
would seem to its human opponent to
have a far less rigid
game personality,
and tricks which would defeat it at an
earlier stage may now fail to deceive
it.
The present level of these learning
machines is that they play a fair
amateur game at chess but that in
checkers they can show a marked
superiority to the player who has
programmed them after from
10
to
20
playing hours of working and in-
doctrination. They thus most definite-
ly escape from the completely effective
control of the man who has made
them. Rigid as the repertory of factors
may be which they are in a position to
take into consideration, they do un-
questionably-and so say those who
have played with them-show original-
ity, not merely in their tactics, which
may be quite unforeseen, but even in
the detailed weighting of their strategy.
As I have said, checker-playing ma-
chines which learn have developed to
the point at which they can defeat
the programmer. However, they ap-
pear still to have one weakness. This
lies in the end game. Here the ma-
chines are somewhat clumsy in de-
termining the best way to give the
coup
de
grbce.
This is due to the fact
that the existing machines have for
the most part adopted a program in
SCIENCE,
VOL.
131