the solution is not a single imputation but rather a system of
imputations.

1.2.1. Although the concept of solution that is used in the
oligopoly solution proposed in Chapters IV through VI is akin
to game theory, the methods used in deriving the solution are
different. Thus far the development in the Theory of Games
has been centered in the solution of zero-sum games (where the
losses of some persons in the game are equivalent to the gains of
the remaining persons) and the min-max games (where with two
players, though both want to maximize their gains, because it is
a zero-sum game one player must have negative gains so that
instead of maximizing his gains he is minimizing his losses). ³ In
economics, other situations are encountered, namely, non-zero-
sum and max-max games. In Sec. 13.3.5, Stackelberg's solution
is expressed as a max-max game solution. A more satisfactory
solution for non-zero-sum games and max-max type solutions is
still needed. ⁴

1.2.2. We use the indifference curves technique in the case
of two sellers and the results are extended to the case of n sellers.
The proofs that are given are far from rigorous but, in presenting
these problems, it is hoped that the attention of mathematicians
will be attracted to some of the peculiar problems of economics
and that as a result new mathematical techniques will be de-
veloped to cope with the special problems.

1.2.3. In the proposed oligopoly solution, rather than giving
definite sets of profits or prices for the sellers, we shall be con-
cerned with directions only, that is, with whether the sellers
will decide to increase or decrease their prices. It is to be noted
here that although theories of competition or monopoly give
definite points of solutions in mathematical or graphic form, in
actual practice definite prices and profits cannot be determined
with this procedure because demand curves, supply curves,

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