Let　r　＞=　2　and　(X-i,G)　(i=1,center　dot　center　dot　center　dot,r)　be　topological　dynamical　systems　with　an　infinite　countable　discrete　amenable　phase　group　G.　Suppose　that　pi(i):(X-i,G)-＞(Xi+1,G)　are　factor　maps,　a={a(1),center　dot　center　dot　center　dot,a(r-1)}is　an　element　of　Rr-1　is　a　vector　with　0　＜=　a(i)　＜=　1　and　(w(1),center　dot　center　dot　center　dot,w(r))is　an　element　of　R-r　is　a　probability　vector　associated　with　a.　In　this　paper,　given　f　is　an　element　of　C(X1),　we　introduce　the　weighted　topological　pressure　P-a(f,G).　Moreover,　by　using　measure-theoretical　theory,　we　establish　a　variational　principle:　P-a(f,G)=sup(mu　is　an　element　of　M)(G)((X)1)(Sigma(r)(i=1)w(i)h(mu)i(X-i,G)+w(1)integral(X)1fd　mu),　where　h({center　dot})(center　dot,G)　is　the　Kolmogorov-Sinai　entropy　of　the　systems　acted　by　the　amenable　group　G　and　mu(i)=pi(i-1)circle　center　dot　center　dot　center　dot　circle　pi(1)mu　is　the　induced　G-invariant　measure　on　X-i.