This paper investigates the problem of constructing predicted estimates for dynamic systems described by difference equations under uncertainty. In particular, the Richardson Arms Model is considered for two competing countries, which can be used to describe the dynamics of their armaments. The main attention is paid to the case when the armament level of one of the countries under consideration is partially unknown. The function describing this level belongs to given compact and convex sets, while the values of the system variables are known only at certain discrete moments of time. An approach is proposed to find the optimal strategy of the first country, which guarantees the achievement of a given result with the maximum uncertainty value with respect to the second country. Analytical relations for determining the maximum values of uncertain quantities are obtained and substantiated. The developed approach is based on the use of properties of convex sets, vector representations, and methods of analysis of dynamic systems. A number of statements and their consequences are proved, which ensure the efficient calculation of estimates in the presence of additional restrictions on the system parameters. The proposed approach can be applied to modeling and analyzing processes of population dynamics, information dissemination, and other socio-economic systems in conditions of incomplete information regarding the state of the system.