nullsum

Nullsum is a term used to describe situations in which the total sum of quantities across participants is zero. In many sources, nullsum is used interchangeably with zero-sum, though in formal mathematics and game theory the phrase zero-sum is more standard and precise.

In game theory, a nullsum (zero-sum) game is one in which any gain by one player is exactly balanced by a loss by the others; the sum of all players’ payoffs is zero for every outcome. This implies that the players compete over a fixed amount of value, with no net creation or destruction of resources within the game.

Examples include two-player matrix games where the payoff to player A is p and the payoff to player B is -p, ensuring that p plus (-p) equals zero for each strategy pair. The concept extends to multiplayer settings, where the sum of all participants’ payoffs remains zero regardless of strategy.

Beyond strictly strategic contexts, nullsum can describe any collection of financial, numerical, or probabilistic components whose net total is zero when all relevant inflows and outflows are accounted for. In such uses, it emphasizes balance rather than wealth creation.

Nullsum is distinct from constant-sum situations, where the total payoff is fixed but not necessarily zero, and from nonzero-sum settings, where the total payoff can vary. It should also not be confused with the null (empty) set in set theory. See also zero-sum game, nonzero-sum game, and constant-sum game.