 This result, as an aside, makes it clear why the team semantics of dependence logic satisfies the empty set property and the downwards closure property. Indeed, if \(X = \emptyset\) then every strategy for Player \(0\) in \(G_{M, X}(\phi)\) is trivially winning and uniform; and if \(X \subseteq Y\) then any uniform winning strategy for Player \(0\) in \(G_{M, X}(\phi)\) is also a uniform winning strategy for Player \(0\) in \(G_{M, Y}(\phi)\).