Constant-sum game ($u_1+u_2=K$)

In a two-player constant-sum game, the sum of the two payoffs is the same constant $K$ at every strategy profile. Replacing each player's payoff by $u_i-K/2$ gives a zero-sum game without changing best responses or Nash equilibria. If both players can guarantee $K/2$, then any Nash equilibrium gives each that value. An equilibrium in pure strategies must use pure strategies with that guarantee: otherwise the opponent could improve on $K/2$ by a unilateral deviation.