Prediction with expert advice is one of the most fundamental problems in online learning. In its simplest form, a learner must choose from one of K actions (possibly choosing a randomized mixture of actions) every round for T rounds. An adversary then reveals a loss vector containing the loss for each action to the learner, the learner incurs their appropriate loss, and play proceeds to the next round. The goal of the learner in such settings is usually to minimize their regret: the gap between their total utility at the end of the game and the maximum utility they could have received if they played the best fixed action in hindsight. Notably, it is possible to construct algorithms for the learner which achieve regret sublinear in T against any adversarially chosen sequence of losses. Traditionally, a learner may use the entire history of losses up until round t to decide their action at round t. In this paper, we investigate the question of what happens if we restrict the learner's action at time t to depend on the losses in some subset of these rounds. Formally, we require the learner's (randomized) action at time t to be a function of the losses in some subset of rounds S

The model checking problem for multi-agent systems against specifications in the alternating-time temporal logic ATL, hence ATL∗, under perfect recall and imperfect information is known to be undecidable. To tackle this problem, in this paper we investigate a notion of bounded recall under incomplete information. We present a novel three-valued semantics for ATL∗ in this setting and analyse the corresponding model checking problem. We show that the three-valued semantics here introduced is an approximation of the classic two-valued semantics, then give a sound, albeit partial, algorithm for model checking two-valued perfect recall via its approximation as three-valued bounded recall. Finally, we extend MCMAS, an open-source model checker for ATL and other agent specifications, to incorporate bounded recall; we illustrate its use and present experimental results.