On the monotonicity of an optimal choice with respect to a parameter

The optimization and game models with ordered strategies in which the optimal choice increases monotonically is studied. An order relation is considered on a universal set and the preferences of a choice-making agent are described. A correspondence from a poset to a relatively complete lattice has a monotone selection if the correspondence increases and each value of the correspondence is complete in the complete lattice. The results also show that if a parametric representation is strongly regular, then the existence of a monotone selection for any correspondence is equivalent to the monotonicity. The parametric representations of preferences that ensure the monotonicity for all sublattices cannot be characterized by simple conditions. For a relatively complete lattice and a parametric representation, a strongly regular and monotone is found to exists and any correspondence is found to possess a monotone selection.