This article introduces a teacher-student framework for reinforcement learning, synthesising and extending material that appeared in conference proceedings [Torrey, L., & Taylor, M. E. (2013)]. Teaching on a budget: Agents advising agents in reinforcement learning. Proceedings of the international conference on autonomous agents and multiagent systems] and in a non-archival workshop paper [Carboni, N., & Taylor, M. E. (2013, May)]. Preliminary results for 1 vs. 1 tactics in StarCraft. Proceedings of the adaptive and learning agents workshop (at AAMAS-13)]. In this framework, a teacher agent instructs a student agent by suggesting actions the student should take as it learns. However, the teacher may only give such advice a limited number of times. We present several novel algorithms that teachers can use to budget their advice effectively, and we evaluate them in two complex video games: StarCraft and Pac-Man. Our results show that the same amount of advice, given at different moments, can have different effects on student learning, and that teachers can significantly affect student learning even when students use different learning methods and state representations.