Zero-sum

From Wikipedia, the free encyclopedia.

Zero-sum describes a situation in which a participant's gain (or loss) is exactly balanced by the losses (or gains) of the other participant(s). It is so named because when you add up the total gains of the participants and subtract the total losses then they will sum to zero. Cutting a cake is zero- or constant-sum because taking a larger piece for yourself reduces the amount of cake available for others. Situations where participants can all gain or suffer together, such as a country with an excess of bananas trading with an other country for their excess of apples where both benefit from the transaction, are referred to as non-zero-sum.

The concept was first developed in game theory and consequently zero-sum situations are often called zero-sum games though this does not imply that the concept, or game theory itself, applies only to what are commonly referred to as games. Optimal strategies for two-player zero-sum games can often be found using minimax strategies.

In 1944 John von Neumann and Oskar Morgenstern proved that any zero-sum game involving n players is in fact a generalised form of a zero-sum game for two persons; and that any non-zero-sum game for n players can be reduced to a zero-sum game for n + 1 players, the (n + 1) th player representing the global profit or loss.

This means that the zero-sum game for two players forms the essential core of mathematical game theory.

(The two paragraphs above are translated from the French article on zero-sum games)

To treat a non-zero-sum situation as a zero-sum situation, or to believe that all situations are zero-sum situations, is called the zero-sum fallacy.

(Italics added by me.))