In this paper, the formation control problem of a multi-agent system is studied. The foraging behavior is modeled as a finite-horizon non-cooperative differential game under local information, and the existence and properties of Nash equilibria are studied. The formations are achieved in an intrinsic way in the sense that they are only attributed to the inter-agent interaction and geometric properties of the network, where the desired formations are not designated beforehand. Through the design of individual costs and network topology, regular polygons, antipodal formations and Platonic solids are achieved as Nash equilibria while inter-agent collision is avoided. While the focus is on the finite horizon case, it is also studied how the formation patterns would change as the length of the time interval tends to infinity. Finally, numerical simulations are provided in both two-dimensional and three-dimensional Euclidean space to demonstrate the effectiveness and feasibility of the proposed methods.