We consider a class of $0$-$1$ polynomial programming termed multiple choice polynomial programming (MCPP) where the constraint requires exact one component per subset of the partition to be $1$ after all the entries are partitioned. Compared to the unconstrained counterpart, there are few polyhedral studies of MCPP in general form. This paper serves as the first attempt to propose a polytope associated with a hypergraph to study MCPP, which is the convex hull of $0$-$1$ vectors satisfying multiple choice constraints and production constraints. With the help of the decomposability property, we obtain an explicit half-space representation of the MCPP polytope when the underlying hypergraph is $α$-acyclic by induction on the number of hyperedges, which is an analogy of the acyclicity results on the multilinear polytope by Del Pia and Khajavirad (SIAM J Optim 28 (2018) 1049) when the hypergraph is $γ$-acyclic. We also present a necessary and sufficient condition for the inequalities lifted from the facet-inducing ones for the multilinear polytope to be still facet-inducing for the MCPP polytope. This result covers the particular cases by Bärmann, Martin and Schneider (SIAM J Optim 33 (2023) 2909).