 For every vocabulary \(\tau\), \(\mathbf{IFL}[\tau]\) formula \(\phi\), model \((\tau\) structure) \(M\), and variable assignment \(g\), a two-player zero-sum game \(G(\phi , M, g)\) between player 1 and player 2 is  associated.[20]  If \(g\) is a variable assignment, \(g[x/a]\) is the variable assignment which is otherwise like \(g\) but maps the variable \(x\) to the object \(a\).