One of the best known ideas in the study of bounded rationality is Simon's satisficing; yet we still lack a standard formalization of the heuristic and its implications. We propose a mathematical model of satisficing which explicitly represents agents' aspirations and which explores both single-person and multi-player contexts. The model shows that satisficing has a signature performance-profile in both contexts: (1) it can induce optimal long-run behavior in one class of problems but not in the complementary class; and (2) in the latter, it generates behavior that is sensible but not optimal. The model also yields empirically testable predictions: in certain bandit-problems it pins down the limiting probabilities of each arm's use, and it provides an ordering of the arms' dynamical use-probabilities as well.