Suppose a couple is getting divorced and they want to split some collection of objetcs, say their books. They have $N$ books, labeled with indices from $1$ to $N$.
You are the arbiter of this division. In order to do it you require that the husband gives you a set of preferences, $\{h_1,...,h_N\}$, with $0\le h_i\le 1$ being a measure of how much he wants to have book $i$. Likewise $\{w_1,...,w_N\}$ for the wife. The preferences satisfy the normalization $$\sum_{i=1}^Nh_i=\sum_{i=1}^Nw_i=1.$$
Let $A\subset\{1,...,N\}$ be the set containing the indices of the books that are assigned to the husband, while its complement $B=\{1,...,N\}\setminus A$ is associated with the wife. The satisfaction of the husband with the arrangement is $$H=\sum_{i\in A}h_i,$$ while the satisfaction of the wife is $$W=\sum_{j\in B}w_j.$$
The total satisfaction of the couple is the product $S=HW$. Your job is to maximize $S$ with respect to the set partition $(A,B)$.
Is this a classical game? Has it been studied? Does it have a name?
Not sure if it has been studied or if it has a name, but you can solve the problem via quadratic unconstrained binary optimization (QUBO) as follows. Let binary decision variable $x_i$ indicate whether $i\in A$. The problem is to maximize $$H W = \left(\sum_{i=1}^N h_i x_i\right) \left(\sum_{j=1}^N w_j (1-x_j)\right)= \sum_{i=1}^N \sum_{j=1}^N h_i w_j x_i (1-x_j).$$
Alternatively, you can linearize the objective by introducing $y_{i,j} \ge 0$ to represent the product $x_i x_j$ and maximizing $$\sum_{i=1}^N \sum_{j=1}^N h_i w_j (x_i -y_{i,j})$$ subject to linear constraints $$y_{i,j} \ge x_i + x_j - 1.$$