Formation control deals with the design of decentralized feedback laws that stabilize agents around a target configuration. In this context, the target configuration is a global design variable for the system. We call decentralized design with localized objective the design of control laws in which agents have an incomplete knowledge of the global design variable. We explain how objective localization allows us to address privacy and modularity issues in decentralized systems. Our goal is to develop a systematic approach to establish lower bounds on the amount of information about the global design variable agents need in order to cooperate successfully. To this end, we study a simple four agents formation and show that exponential stabilization is possible with a decentralized execution and non-localized objective but impossible with a decentralized execution and localized objective.