Given a weight vector $w\in [0,1]^d$ such that $\sum w_i=1$, the game goes as follows:

Two players, $X,Y$ choose strategies $x,y\in [0,1]^d$ such that $\sum x_i = \sum y_i = 1$.

The utility (profit) for player $X$ is given by $$u_X=\sum_{i:x_i>y_i} x_i\cdot w_i$$

That is summing over all coordinates in which $x$ is larger than $y$, and the profit for the coordinate is given by $x_i\cdot w_i$.

Y's utility is symmetric ($u_Y=\sum_{i:y_i>x_i} y_i\cdot w_i$).

This also interests me for low dimension (say $d=3$) if it makes it easier.

We assume w.l.o.g that $i>j\implies w_i\geq w_j$.

If $w_1\geq \frac{1}{2}$ then this is trivial, so assume this is not the case.

If needed we may assume that if $x_i=y_i$ then they both get $\frac{x_i\cdot w_i}{2}$ utility for that coordinate.

Is there a known name for this game?

Is the equilibrium of this game computable or is it $PPAD$-hard?