Summary: Let \(A\) be a set of variants and let \(r_1, r_2, \dots , r_k\) be binary relations on \(A\). A choice model of depth \(k\) with sequential decision-making takes each \(X\subseteq A\) into its subset \(C_{r_k}(\dots C_{r_2}(C_{r_1}(X))\dots)\), where \(C_r(Y)=\{ y\in Y\mid (\forall z\in Y)\;z \bar{r} y\}\), \(Y\subseteq A\). The minimization problem consists in finding an equivalent model of the smallest depth. The compression problem is formulated similarly, but the model to be found has to satisfy some ``embedding'' condition. It is proven that, for \(k\geq 3\), both the minimization problem and the compression problem are NP-hard. Parameters of local algorithms for solving these problems are studied. It is shown that the compression problem may be solved by algorithms which operate with neighborhoods of size 3, while there is no finite \(p\) such that the minimization problem can be solved with neighborhoods of size \(p\). For an arbitrary \(k\), a model of depth \(k\) is constructed whose minimization cannot be done if only the neighborhoods different from the whole model are used.