<p>The structure of finite games with symmetric potential functions is investigated in this paper. First, by constructing a basis of symmetric functions, a necessary and sufficient condition is presented to verify whether a finite game has symmetric potential functions. Then, a basis of the subspace of finite games with symmetric potential functions is provided. Next, the symmetric potential game is studied. By proving the symmetry of the potential function, a linear system is also presented for the verification of symmetric potential games, as well as a basis. Finally, as an application of the obtained results, the optimization of quasi-symmetric spatial games is considered. A sufficient condition for the utility design is given to turn the spatial game into a weighted potential game with the preassigned objective function as the potential function.</p>