Zero-sum asymmetric information games model decision making scenarios involving two competing players whohave different information about the game being played. A particular case is that of nested information, where one (informed) player has superior information over the other (uninformed) player. This paper considers the case of nested information in repeated zero-sum games and studies the computation of strategies for both the informed and uninformed players for finite-horizon and discounted infinite-horizon nested information games. For finite-horizon settings, we exploit that for both players, the security strategy, and also the opponent's corresponding best response, depend only on the informed player's history of actions. Using this property, we formulate an LP computation of player strategies that is linear in the size of the uninformed player's action set. For the infinite-horizon discounted game, we construct LP formulations to compute the approximated security strategies for both players, and show that the worst case performance difference between the approximated security strategies and the security strategies converges to zero exponentially. Finally, we illustrate the results on a network interdiction game between an informed system administrator and an uniformed intruder.