Nash equilibrium strategies of general linear-quadratic two-player difference games with two kinds of periodic information structures are considered. Solution algorithms are developed for problems where the players' information is of periodic open-loop or periodic open-closed type. In the former case the players receive measurements of the state periodically only at the beginning of each interval and in the latter case one of the players has a perfect memory information of the state within each period. The solutions are obtained by recursive algorithms where a series of coupled difference equations of Riccati type are solved repeatedly and where the boundary values of these equations are determined by similar difference equations. A new game theoretic worst case design method based on games with periodic open-closed information structure is then proposed and applied to the design of a state regulator for a pilot process. The results obtained in the example suggest that this new approach can be successfully employed in practical design problems.