= Zero-sum game
{title2=$\sum_i u_i(s)=0$}
{wiki}
In a <zero-sum game>, the players' <payoffs> sum to zero at every <pure strategy> <strategy profile>. For two players this means $u_2=-u_1$: maximizing the column player's <payoff> is minimizing the row player's. A single <payoff matrix> $A$ therefore specifies a finite two-player <matrix game>, and <independent> <mixed strategies> $x,y$ give <expected payoffs> $x^{\mathsf T}Ay$ and $-x^{\mathsf T}Ay$.
<Matching pennies> and the usual <rock paper scissors> are two-player <zero-sum games>. The <asymmetric coordination example> is not: its total <payoff> is $5$ when choices match and $0$ otherwise. Applying a two-player <minimax theorem> to general <bimatrix games> would discard this possibility of shared gains. For a two-player <zero-sum game>, however, <optimal mixed strategies> guarantee the <value of a zero-sum game>, and pairing any two opposing optimal strategies gives a <Nash equilibrium>.
Back to article page
