Consider a game between two players, $P$ (row player, maximizer) and $Q$ (column player, minimizer). Player $P$ controls a probability distribution $p = \{p_1, \dots, p_n\}$ and similarly player $Q$ controls a probability distribution $q = \{q_1, \dots, q_n\}$. There is a function $S = \sum_{i = 1}^n p_i \times q_i$. Player $P$ aims to choose a probability distribution $p$ to maximize $S$ and in contrast, player $Q$ aims to select a probability distribution $q$ to minimize $S$. Assuming each player knows the probability distribution of the other player, the optimal strategy (NE mixed strategy) for both players is the uniform distribution. That is the optimal strategy is: $p_i = q_i = 1/n, \forall i, 1 \leq i \leq n$ which results in $S = 1/n$.
I wonder how can I give a better explanation (or proof) that this is the best strategy. I expect there should be an easier explanation/proof than considering it as a zero-sum game and using the Maxmin method.