Do they refer to constituent of sets as "players" instead of "elements" now?

Try various relations on sets with 0, 1, 2, and 3 elements. Consider, for example, antisymmetry: for all x and y, if R(x,y) and R(y,x), then x = y. Note that if it is not the case that R(x,y) and R(y,x) for some x, y, this fact does not violate antisymmetry because an implication is vacuously true when the assumption is false. So, to break antisymmetry, you need to have x and y such that the assumption is true but the conclusion is false.