<Dd> r i (σ − i) = arg ⁡ max σ i u i (σ i, σ − i) (\ displaystyle r_ (i) (\ sigma _ (- i)) = \ arg \ max _ (\ sigma _ (i)) u_ (i) (\ sigma _ (i), \ sigma _ (- i))) </Dd> <P> Here, σ ∈ Σ (\ displaystyle \ sigma \ in \ Sigma), where Σ = Σ i × Σ − i (\ displaystyle \ Sigma = \ Sigma _ (i) \ times \ Sigma _ (- i)), is a mixed - strategy profile in the set of all mixed strategies and u i (\ displaystyle u_ (i)) is the payoff function for player i . Define a set - valued function r: Σ → 2 Σ (\ displaystyle r \ colon \ Sigma \ rightarrow 2 ^ (\ Sigma)) such that r = (r i (σ − i), r − i (σ i)) (\ displaystyle r = (r_ (i) (\ sigma _ (- i)), r_ (- i) (\ sigma _ (i)))). The existence of a Nash equilibrium is equivalent to r (\ displaystyle r) having a fixed point . </P> <P> Kakutani's fixed point theorem guarantees the existence of a fixed point if the following four conditions are satisfied . </P> <Ol> <Li> Σ (\ displaystyle \ Sigma) is compact, convex, and nonempty . </Li> <Li> r (σ) (\ displaystyle r (\ sigma)) is nonempty . </Li> <Li> r (σ) (\ displaystyle r (\ sigma)) is convex . </Li> <Li> r (σ) (\ displaystyle r (\ sigma)) is upper hemicontinuous </Li> </Ol>

Another name for the combination strategy is multiple strategies