Characterization of countable and continuous Richter–Peleg multi-utility representations

This paper contributes to the theoretical literature on decision models where agents may encounter challenges in comparing alternatives. We introduce a characterization of countable Richter–Peleg multi-utility representations, both semicontinuous (upper and lower) and continuous, within preorders that may not be total. The proposed theorems provide a comprehensive mathematical framework, complementing previous results of Alcantud et al. and Bosi on countable multi-utility representations. Our characterizations establish necessary and sufficient conditions through topological properties and constructive methods via indicator functions. Furthermore, we introduce a topological framework aligned with the property of strong local non-satiation and provide a novel theorem containing sufficient conditions for the existence of countable upper semi-continuous multi-utility representations of a preorder. The results demonstrate that preference representations can be achieved using countably many functions rather than uncountable families, with implications for computational tractability and the identification of maximal elements in optimization contexts.